{"id":2892,"date":"2025-06-05T03:13:36","date_gmt":"2025-06-05T03:13:36","guid":{"rendered":"https:\/\/diznr.com\/?p=2892"},"modified":"2025-06-05T03:13:36","modified_gmt":"2025-06-05T03:13:36","slug":"day-03part-04-discrete-mathematics-short-trick-method-for-solving-nested-problems-quantifiers","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-03part-04-discrete-mathematics-short-trick-method-for-solving-nested-problems-quantifiers\/","title":{"rendered":"Day 03Part 04- Discrete mathematics &#8211; Short trick method for solving Nested Quantifiers problems."},"content":{"rendered":"<p>Day 03Part 04- Discrete mathematics &#8211; Short trick method for solving Nested Quantifiers problems.<\/p>\n<p>[fvplayer id=&#8221;161&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"52\"><strong data-start=\"4\" data-end=\"50\">\u00a0Day 03 | Part 04 &#8211; Discrete Mathematics<\/strong><\/h3>\n<h3 data-start=\"53\" data-end=\"121\"><strong data-start=\"56\" data-end=\"119\">Short Trick Methods for Solving Nested Quantifiers Problems<\/strong><\/h3>\n<p data-start=\"123\" data-end=\"358\">Nested quantifiers can be tricky, but with <strong data-start=\"166\" data-end=\"192\">some simple techniques<\/strong>, you can solve them <strong data-start=\"213\" data-end=\"239\">quickly and accurately<\/strong>. Let&#8217;s go through some <strong data-start=\"263\" data-end=\"291\">short tricks and methods<\/strong> to solve <strong data-start=\"301\" data-end=\"331\">nested quantifier problems<\/strong> in Discrete Mathematics.<\/p>\n<h3 data-start=\"365\" data-end=\"418\"><strong data-start=\"368\" data-end=\"416\">\u00a01. Understanding the Order of Quantifiers<\/strong><\/h3>\n<p data-start=\"419\" data-end=\"515\">The order of <strong data-start=\"432\" data-end=\"460\">\u2200 (Universal Quantifier)<\/strong> and <strong data-start=\"465\" data-end=\"495\">\u2203 (Existential Quantifier)<\/strong> matters in logic.<\/p>\n<p data-start=\"517\" data-end=\"532\"><strong data-start=\"519\" data-end=\"530\">Rule 1:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2200x\u2203yP(x,y)\u2260\u2203y\u2200xP(x,y)\u2200x \u2203y P(x, y) \\neq \u2203y \u2200x P(x, y)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2200<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"inner\"><span class=\"mord\">\ue020<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mord\">\u2200<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"574\" data-end=\"789\">\u00a0If <strong data-start=\"580\" data-end=\"609\">Universal (\u2200) comes first<\/strong>, then for <strong data-start=\"620\" data-end=\"631\">every x<\/strong>, we must find at least one <strong data-start=\"659\" data-end=\"664\">y<\/strong> that satisfies P(x, y).<br data-start=\"688\" data-end=\"691\" \/>If <strong data-start=\"697\" data-end=\"728\">Existential (\u2203) comes first<\/strong>, we must find at least one <strong data-start=\"756\" data-end=\"761\">y<\/strong> that works for <strong data-start=\"777\" data-end=\"786\">all x<\/strong>.<\/p>\n<p data-start=\"791\" data-end=\"808\"><strong data-start=\"793\" data-end=\"806\">Shortcut:<\/strong><\/p>\n<ul data-start=\"809\" data-end=\"974\">\n<li data-start=\"809\" data-end=\"901\">If the <strong data-start=\"818\" data-end=\"846\">same type of quantifiers<\/strong> (\u2200\u2026\u2200 or \u2203\u2026\u2203) appear together, order does NOT matter.<\/li>\n<li data-start=\"902\" data-end=\"974\">If <strong data-start=\"907\" data-end=\"939\">different quantifiers (\u2200, \u2203)<\/strong> appear, <strong data-start=\"948\" data-end=\"965\">order matters<\/strong> a lot!<\/li>\n<\/ul>\n<h3 data-start=\"981\" data-end=\"1043\"><strong data-start=\"984\" data-end=\"1041\">\u00a02. Quick Trick to Solve Nested Quantifier Problems<\/strong><\/h3>\n<h3 data-start=\"1044\" data-end=\"1107\"><strong data-start=\"1048\" data-end=\"1105\">Trick 1: Consider a Simple Example (Use Small Values)<\/strong><\/h3>\n<p data-start=\"1108\" data-end=\"1230\">When solving a nested quantifier problem, try assigning <strong data-start=\"1164\" data-end=\"1180\">small values<\/strong> to variables to check if the statement is true.<\/p>\n<p data-start=\"1232\" data-end=\"1250\"><strong data-start=\"1234\" data-end=\"1248\">Example 1:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2200x\u2203y(x+y=10)\u2200x \u2203y (x + y = 10)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2200<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1278\" data-end=\"1517\"><strong data-start=\"1280\" data-end=\"1291\">Step 1:<\/strong> Pick <strong data-start=\"1297\" data-end=\"1306\">any x<\/strong> (say x = 3)<br data-start=\"1318\" data-end=\"1321\" \/><strong data-start=\"1323\" data-end=\"1334\">Step 2:<\/strong> Find <strong data-start=\"1340\" data-end=\"1350\">some y<\/strong> such that <span class=\"katex\"><span class=\"katex-mathml\">x+y=10x + y = 10<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span><br data-start=\"1377\" data-end=\"1380\" \/><strong data-start=\"1382\" data-end=\"1395\">Solution:<\/strong> If <span class=\"katex\"><span class=\"katex-mathml\">x=3x = 3<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">y=7y = 7<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><\/span><\/span><\/span> works.<br data-start=\"1435\" data-end=\"1439\" \/>\u00a0Since we can always find such a y for every x, the statement is <strong data-start=\"1506\" data-end=\"1514\">true<\/strong>.<\/p>\n<p data-start=\"1519\" data-end=\"1537\"><strong data-start=\"1521\" data-end=\"1535\">Example 2:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2203x\u2200y(x+y=10)\u2203x \u2200y (x + y = 10)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2200<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1565\" data-end=\"1823\"><strong data-start=\"1567\" data-end=\"1578\">Step 1:<\/strong> Find <strong data-start=\"1584\" data-end=\"1594\">some x<\/strong> that works for <strong data-start=\"1610\" data-end=\"1619\">all y<\/strong>.<br data-start=\"1620\" data-end=\"1623\" \/><strong data-start=\"1625\" data-end=\"1636\">Step 2:<\/strong> If <span class=\"katex\"><span class=\"katex-mathml\">x=3x = 3<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">3+y=103 + y = 10<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span> must be true for every y.<br data-start=\"1700\" data-end=\"1703\" \/><strong data-start=\"1705\" data-end=\"1719\">Impossible<\/strong>, because y can take any value, but <span class=\"katex\"><span class=\"katex-mathml\">x+yx + y<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span> should always be 10.<br data-start=\"1787\" data-end=\"1790\" \/>\u00a0This statement is <strong data-start=\"1811\" data-end=\"1820\">false<\/strong>.<\/p>\n<h3 data-start=\"1830\" data-end=\"1883\"><strong data-start=\"1834\" data-end=\"1881\">Trick 2: Swap Quantifiers and Check Meaning<\/strong><\/h3>\n<p data-start=\"1884\" data-end=\"1972\">If a question seems difficult, <strong data-start=\"1915\" data-end=\"1923\">swap<\/strong> quantifiers and check how the meaning changes.<\/p>\n<p data-start=\"1974\" data-end=\"1990\"><strong data-start=\"1976\" data-end=\"1988\">Example:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2200x\u2203yP(x,y)\u2200x \u2203y P(x, y) <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2200<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2014\" data-end=\"2216\">\u00a0Means: &#8220;For every x, there exists some y such that P(x, y) is true.&#8221;<br data-start=\"2084\" data-end=\"2087\" \/><strong data-start=\"2090\" data-end=\"2113\">This is weaker than<\/strong> <span class=\"katex\"><span class=\"katex-mathml\">\u2203y\u2200xP(x,y)\u2203y \u2200x P(x, y)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mord\">\u2200<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>, because in the second case, we must find <strong data-start=\"2176\" data-end=\"2192\">one single y<\/strong> that works for all x.<\/p>\n<h3 data-start=\"2223\" data-end=\"2282\"><strong data-start=\"2227\" data-end=\"2280\">Trick 3: Use Counterexamples for False Statements<\/strong><\/h3>\n<p data-start=\"2283\" data-end=\"2355\">If you suspect a statement is false, try to find a <strong data-start=\"2334\" data-end=\"2352\">counterexample<\/strong>.<\/p>\n<p data-start=\"2357\" data-end=\"2373\"><strong data-start=\"2359\" data-end=\"2371\">Example:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2203x\u2200y(x\u2212y&gt;0)\u2203x \u2200y (x &#8211; y &gt; 0)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2200<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2400\" data-end=\"2643\">\u00a0This means: &#8220;There exists some x such that for all y, <span class=\"katex\"><span class=\"katex-mathml\">x\u2212yx &#8211; y<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span> is always positive.&#8221;<br data-start=\"2488\" data-end=\"2491\" \/>\u00a0But if we choose <span class=\"katex\"><span class=\"katex-mathml\">x=5x = 5<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span>, and y can be <strong data-start=\"2536\" data-end=\"2550\">any number<\/strong>, what if <span class=\"katex\"><span class=\"katex-mathml\">y=10y = 10<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span>?<br data-start=\"2573\" data-end=\"2576\" \/><span class=\"katex\"><span class=\"katex-mathml\">5\u221210=\u221255 &#8211; 10 = -5<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span> (not positive) \u2192 This statement is <strong data-start=\"2631\" data-end=\"2640\">false<\/strong>.<\/p>\n<p data-start=\"2645\" data-end=\"2744\"><strong data-start=\"2648\" data-end=\"2663\">Conclusion:<\/strong> If you can find even <strong data-start=\"2685\" data-end=\"2707\">one counterexample<\/strong>, the whole statement is <strong data-start=\"2732\" data-end=\"2741\">false<\/strong>!<\/p>\n<h3 data-start=\"2751\" data-end=\"2812\"><strong data-start=\"2755\" data-end=\"2810\">Trick 4: Convert Quantified Statements into English<\/strong><\/h3>\n<p data-start=\"2813\" data-end=\"2903\">Sometimes, converting statements into <strong data-start=\"2851\" data-end=\"2868\">plain English<\/strong> makes them easier to understand.<\/p>\n<p data-start=\"2905\" data-end=\"2921\"><strong data-start=\"2907\" data-end=\"2919\">Example:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2200x\u2203y(x\u22c5y=1)\u2200x \u2203y (x \\cdot y = 1)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2200<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2203<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u22c5<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2952\" data-end=\"3165\">\u00a0Means: &#8220;For every x, there exists some y such that their product is 1.&#8221;<br data-start=\"3025\" data-end=\"3028\" \/>\u00a0If <span class=\"katex\"><span class=\"katex-mathml\">x=0x = 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">x\u22c5y=1x \\cdot y = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u22c5<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> is <strong data-start=\"3074\" data-end=\"3088\">impossible<\/strong> (because <span class=\"katex\"><span class=\"katex-mathml\">0\u00d7y=00 \\times y = 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span>, not 1).<br data-start=\"3127\" data-end=\"3130\" \/>\u00a0So this statement is <strong data-start=\"3153\" data-end=\"3162\">false<\/strong>.<\/p>\n<h3 data-start=\"3172\" data-end=\"3225\"><strong data-start=\"3175\" data-end=\"3223\">\u00a03. Quick Recap \u2013 Problem-Solving Strategy<\/strong><\/h3>\n<p data-start=\"3226\" data-end=\"3621\"><strong data-start=\"3228\" data-end=\"3239\">Step 1:<\/strong> Identify <strong data-start=\"3249\" data-end=\"3258\">order<\/strong> of quantifiers (Universal or Existential).<br data-start=\"3301\" data-end=\"3304\" \/><strong data-start=\"3306\" data-end=\"3317\">Step 2:<\/strong> Pick <strong data-start=\"3323\" data-end=\"3340\">simple values<\/strong> to test if the statement holds.<br data-start=\"3372\" data-end=\"3375\" \/><strong data-start=\"3377\" data-end=\"3388\">Step 3:<\/strong> Try <strong data-start=\"3393\" data-end=\"3417\">swapping quantifiers<\/strong> to see how the meaning changes.<br data-start=\"3449\" data-end=\"3452\" \/><strong data-start=\"3454\" data-end=\"3465\">Step 4:<\/strong> Look for <strong data-start=\"3475\" data-end=\"3494\">counterexamples<\/strong> to disprove false statements.<br data-start=\"3524\" data-end=\"3527\" \/><strong data-start=\"3529\" data-end=\"3540\">Step 5:<\/strong> Convert the logical statement into <strong data-start=\"3576\" data-end=\"3593\">plain English<\/strong> for better understanding.<\/p>\n<h3 data-start=\"3628\" data-end=\"3650\"><strong data-start=\"3631\" data-end=\"3648\">\u00a0Conclusion<\/strong><\/h3>\n<p data-start=\"3651\" data-end=\"3932\"><strong data-start=\"3654\" data-end=\"3676\">Nested quantifiers<\/strong> are essential for logical reasoning in computer science.<br data-start=\"3733\" data-end=\"3736\" \/>\u00a0Using <strong data-start=\"3745\" data-end=\"3802\">small values, counterexamples, and translation tricks<\/strong>, we can <strong data-start=\"3811\" data-end=\"3836\">solve problems faster<\/strong>.<br data-start=\"3837\" data-end=\"3840\" \/><strong data-start=\"3843\" data-end=\"3872\">Changing quantifier order<\/strong> significantly impacts the meaning of a logical statement.<\/p>\n<p data-start=\"3939\" data-end=\"4010\" data-is-last-node=\"\" data-is-only-node=\"\">\u00a0<strong data-start=\"3942\" data-end=\"4007\">Do you want more practice problems or examples on this topic?<\/strong><\/p>\n<h3 data-start=\"3939\" data-end=\"4010\"><a href=\"https:\/\/www.eecs.yorku.ca\/course_archive\/2008-09\/S\/1019\/Website_files\/06-nested-quantifiers.pdf\" target=\"_blank\" rel=\"noopener\">Day 03Part 04- Discrete mathematics &#8211; Short trick method for solving Nested Quantifiers problems.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.math.uh.edu\/~irina\/MATH3336\/3336Notes\/3336S15.pdf\" target=\"_blank\" rel=\"noopener\">Nested Quantifiers<\/a><\/h3>\n<p data-start=\"0\" data-end=\"40\">Absolutely! Here&#8217;s your requested guide:<\/p>\n<hr data-start=\"42\" data-end=\"45\" \/>\n<h1 data-start=\"47\" data-end=\"96\">\ud83d\udcd8 <strong data-start=\"52\" data-end=\"94\">Day 03 \u2013 Part 04: Discrete Mathematics<\/strong><\/h1>\n<h2 data-start=\"97\" data-end=\"167\">\ud83e\udde0 <strong data-start=\"103\" data-end=\"165\">Short Trick Method for Solving Nested Quantifiers Problems<\/strong><\/h2>\n<p data-start=\"168\" data-end=\"217\"><em data-start=\"168\" data-end=\"217\">(Useful for GATE, CS, CBSE, and BTech students)<\/em><\/p>\n<hr data-start=\"219\" data-end=\"222\" \/>\n<h2 data-start=\"224\" data-end=\"262\">\ud83d\udd0d <strong data-start=\"230\" data-end=\"262\">What are Nested Quantifiers?<\/strong><\/h2>\n<p data-start=\"263\" data-end=\"327\">Nested quantifiers are <strong data-start=\"286\" data-end=\"320\">quantifiers within quantifiers<\/strong>, like:<\/p>\n<ul data-start=\"329\" data-end=\"366\">\n<li data-start=\"329\" data-end=\"348\">\n<p data-start=\"331\" data-end=\"348\"><code data-start=\"331\" data-end=\"346\">\u2200x \u2203y P(x, y)<\/code><\/p>\n<\/li>\n<li data-start=\"349\" data-end=\"366\">\n<p data-start=\"351\" data-end=\"366\"><code data-start=\"351\" data-end=\"366\">\u2203x \u2200y P(x, y)<\/code><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"368\" data-end=\"477\">These are <strong data-start=\"378\" data-end=\"415\">multi-variable logical statements<\/strong>, and the <strong data-start=\"425\" data-end=\"449\">order of quantifiers<\/strong> changes the entire meaning!<\/p>\n<hr data-start=\"479\" data-end=\"482\" \/>\n<h2 data-start=\"484\" data-end=\"546\">\u2705 <strong data-start=\"489\" data-end=\"546\">Shortcut Methods to Solve Nested Quantifier Questions<\/strong><\/h2>\n<h3 data-start=\"548\" data-end=\"600\">\ud83d\udca1 <strong data-start=\"555\" data-end=\"600\">TRICK #1: Start from Innermost Quantifier<\/strong><\/h3>\n<p data-start=\"601\" data-end=\"710\">Always <strong data-start=\"608\" data-end=\"633\">solve from inside out<\/strong>. Work with the innermost quantifier first.<br data-start=\"676\" data-end=\"679\" \/>Break it down <strong data-start=\"693\" data-end=\"709\">step by step<\/strong>.<\/p>\n<hr data-start=\"712\" data-end=\"715\" \/>\n<h3 data-start=\"717\" data-end=\"783\">\ud83d\udca1 <strong data-start=\"724\" data-end=\"783\">TRICK #2: Use Small Finite Domains for Quick Evaluation<\/strong><\/h3>\n<p data-start=\"784\" data-end=\"879\">If not mentioned, <strong data-start=\"802\" data-end=\"827\">assume a small domain<\/strong>, like {1, 2} or {0, 1} to manually check the logic.<\/p>\n<h3 data-start=\"881\" data-end=\"893\">Example:<\/h3>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none rounded-t-[5px]\">text<\/div>\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-sidebar-surface-primary text-token-text-secondary dark:bg-token-main-surface-secondary flex items-center rounded-sm px-2 font-sans text-xs\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"Copy\">Copy<\/button><span class=\"\" data-state=\"closed\"><button class=\"flex items-center gap-1 px-4 py-1 select-none\">Edit<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre! language-text\">\u2200x \u2203y (x + y = 3) over domain {1, 2}<br \/>\n<\/code><\/div>\n<\/div>\n<ul data-start=\"944\" data-end=\"1012\">\n<li data-start=\"944\" data-end=\"967\">\n<p data-start=\"946\" data-end=\"967\">x = 1 \u2192 \u2203y: y = 2 \u2705<\/p>\n<\/li>\n<li data-start=\"968\" data-end=\"1012\">\n<p data-start=\"970\" data-end=\"1012\">x = 2 \u2192 \u2203y: y = 1 \u2705<br data-start=\"989\" data-end=\"992\" \/>\u2705 Statement is TRUE.<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1014\" data-end=\"1017\" \/>\n<h3 data-start=\"1019\" data-end=\"1068\">\ud83d\udca1 <strong data-start=\"1026\" data-end=\"1068\">TRICK #3: Think in Real-life Sentences<\/strong><\/h3>\n<p data-start=\"1069\" data-end=\"1108\">Translate logic into everyday language.<\/p>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"1110\" data-end=\"1269\">\n<thead data-start=\"1110\" data-end=\"1136\">\n<tr data-start=\"1110\" data-end=\"1136\">\n<th data-start=\"1110\" data-end=\"1125\" data-col-size=\"sm\">Logical Form<\/th>\n<th data-start=\"1125\" data-end=\"1136\" data-col-size=\"sm\">English<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1164\" data-end=\"1269\">\n<tr data-start=\"1164\" data-end=\"1211\">\n<td data-start=\"1164\" data-end=\"1180\" data-col-size=\"sm\">\u2200x \u2203y P(x, y)<\/td>\n<td data-col-size=\"sm\" data-start=\"1180\" data-end=\"1211\">&#8220;For every x, there is a y&#8221;<\/td>\n<\/tr>\n<tr data-start=\"1212\" data-end=\"1269\">\n<td data-start=\"1212\" data-end=\"1228\" data-col-size=\"sm\">\u2203x \u2200y P(x, y)<\/td>\n<td data-col-size=\"sm\" data-start=\"1228\" data-end=\"1269\">&#8220;There is one x that works for all y&#8221;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<blockquote data-start=\"1271\" data-end=\"1314\">\n<p data-start=\"1273\" data-end=\"1314\">\u2757 <strong data-start=\"1275\" data-end=\"1314\">Changing order = Different meaning!<\/strong><\/p>\n<\/blockquote>\n<hr data-start=\"1316\" data-end=\"1319\" \/>\n<h3 data-start=\"1321\" data-end=\"1369\">\ud83d\udca1 <strong data-start=\"1328\" data-end=\"1369\">TRICK #4: Use Tables for Verification<\/strong><\/h3>\n<p data-start=\"1370\" data-end=\"1439\">When domain is small, use a <strong data-start=\"1398\" data-end=\"1413\">truth table<\/strong> to test all combinations.<\/p>\n<p data-start=\"1441\" data-end=\"1503\"><strong data-start=\"1441\" data-end=\"1452\">Example<\/strong>:<br data-start=\"1453\" data-end=\"1456\" \/>Let domain = {1, 2}<br data-start=\"1475\" data-end=\"1478\" \/>Check <code data-start=\"1484\" data-end=\"1503\">\u2200x \u2203y (x \u00d7 y = 2)<\/code><\/p>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"1505\" data-end=\"1598\">\n<thead data-start=\"1505\" data-end=\"1523\">\n<tr data-start=\"1505\" data-end=\"1523\">\n<th data-start=\"1505\" data-end=\"1509\" data-col-size=\"sm\">x<\/th>\n<th data-start=\"1509\" data-end=\"1513\" data-col-size=\"sm\">y<\/th>\n<th data-start=\"1513\" data-end=\"1523\" data-col-size=\"sm\">x\u00d7y=2?<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1543\" data-end=\"1598\">\n<tr data-start=\"1543\" data-end=\"1556\">\n<td data-start=\"1543\" data-end=\"1547\" data-col-size=\"sm\">1<\/td>\n<td data-col-size=\"sm\" data-start=\"1547\" data-end=\"1551\">1<\/td>\n<td data-col-size=\"sm\" data-start=\"1551\" data-end=\"1556\">\u274c<\/td>\n<\/tr>\n<tr data-start=\"1557\" data-end=\"1570\">\n<td data-start=\"1557\" data-end=\"1561\" data-col-size=\"sm\">1<\/td>\n<td data-col-size=\"sm\" data-start=\"1561\" data-end=\"1565\">2<\/td>\n<td data-col-size=\"sm\" data-start=\"1565\" data-end=\"1570\">\u2705<\/td>\n<\/tr>\n<tr data-start=\"1571\" data-end=\"1584\">\n<td data-start=\"1571\" data-end=\"1575\" data-col-size=\"sm\">2<\/td>\n<td data-col-size=\"sm\" data-start=\"1575\" data-end=\"1579\">1<\/td>\n<td data-col-size=\"sm\" data-start=\"1579\" data-end=\"1584\">\u2705<\/td>\n<\/tr>\n<tr data-start=\"1585\" data-end=\"1598\">\n<td data-start=\"1585\" data-end=\"1589\" data-col-size=\"sm\">2<\/td>\n<td data-col-size=\"sm\" data-start=\"1589\" data-end=\"1593\">2<\/td>\n<td data-col-size=\"sm\" data-start=\"1593\" data-end=\"1598\">\u274c<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p data-start=\"1600\" data-end=\"1604\">Now:<\/p>\n<ul data-start=\"1605\" data-end=\"1677\">\n<li data-start=\"1605\" data-end=\"1624\">\n<p data-start=\"1607\" data-end=\"1624\">x = 1 \u2192 y = 2 \u2705<\/p>\n<\/li>\n<li data-start=\"1625\" data-end=\"1677\">\n<p data-start=\"1627\" data-end=\"1677\">x = 2 \u2192 y = 1 \u2705<br data-start=\"1642\" data-end=\"1645\" \/>\u2705 So \u2200x \u2203y (x\u00d7y = 2) is <strong data-start=\"1669\" data-end=\"1677\">True<\/strong><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1679\" data-end=\"1682\" \/>\n<h3 data-start=\"1684\" data-end=\"1743\">\ud83d\udca1 <strong data-start=\"1691\" data-end=\"1743\">TRICK #5: Understand the Role of Quantifier Type<\/strong><\/h3>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"1745\" data-end=\"1971\">\n<thead data-start=\"1745\" data-end=\"1772\">\n<tr data-start=\"1745\" data-end=\"1772\">\n<th data-start=\"1745\" data-end=\"1758\" data-col-size=\"sm\">Quantifier<\/th>\n<th data-start=\"1758\" data-end=\"1765\" data-col-size=\"sm\">Goal<\/th>\n<th data-start=\"1765\" data-end=\"1772\" data-col-size=\"sm\">Tip<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"1801\" data-end=\"1971\">\n<tr data-start=\"1801\" data-end=\"1888\">\n<td data-start=\"1801\" data-end=\"1815\" data-col-size=\"sm\">\u2200 (For all)<\/td>\n<td data-col-size=\"sm\" data-start=\"1815\" data-end=\"1847\">Must hold for <strong data-start=\"1831\" data-end=\"1840\">every<\/strong> value<\/td>\n<td data-col-size=\"sm\" data-start=\"1847\" data-end=\"1888\">Find <strong data-start=\"1854\" data-end=\"1876\">any counterexample<\/strong>, it fails<\/td>\n<\/tr>\n<tr data-start=\"1889\" data-end=\"1971\">\n<td data-start=\"1889\" data-end=\"1902\" data-col-size=\"sm\">\u2203 (Exists)<\/td>\n<td data-col-size=\"sm\" data-start=\"1902\" data-end=\"1935\">Must hold for <strong data-start=\"1918\" data-end=\"1934\">at least one<\/strong><\/td>\n<td data-col-size=\"sm\" data-start=\"1935\" data-end=\"1971\">Find <strong data-start=\"1942\" data-end=\"1959\">one true case<\/strong>, it works<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"1973\" data-end=\"1976\" \/>\n<h2 data-start=\"1978\" data-end=\"2031\">\ud83e\uddea <strong data-start=\"1984\" data-end=\"2031\">Practice Short Questions (with Short Trick)<\/strong><\/h2>\n<h3 data-start=\"2033\" data-end=\"2042\">Q1:<\/h3>\n<p data-start=\"2043\" data-end=\"2173\"><strong data-start=\"2043\" data-end=\"2056\">Statement<\/strong>: <code data-start=\"2058\" data-end=\"2073\">\u2203x \u2200y (x \u2265 y)<\/code> over domain {1, 2}<br data-start=\"2092\" data-end=\"2095\" \/>\u2705 Trick: Is there <strong data-start=\"2113\" data-end=\"2122\">any x<\/strong> such that it&#8217;s <strong data-start=\"2138\" data-end=\"2172\">greater than or equal to all y<\/strong>?<\/p>\n<p data-start=\"2175\" data-end=\"2187\">Try x = 2:<\/p>\n<ul data-start=\"2188\" data-end=\"2218\">\n<li data-start=\"2188\" data-end=\"2199\">\n<p data-start=\"2190\" data-end=\"2199\">2 \u2265 1 \u2705<\/p>\n<\/li>\n<li data-start=\"2200\" data-end=\"2218\">\n<p data-start=\"2202\" data-end=\"2218\">2 \u2265 2 \u2705<br data-start=\"2209\" data-end=\"2212\" \/>\u2705 True<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"2220\" data-end=\"2223\" \/>\n<h3 data-start=\"2225\" data-end=\"2234\">Q2:<\/h3>\n<p data-start=\"2235\" data-end=\"2326\"><strong data-start=\"2235\" data-end=\"2248\">Statement<\/strong>: <code data-start=\"2250\" data-end=\"2269\">\u2200x \u2203y (x \u00d7 y = 1)<\/code> in integers<br data-start=\"2281\" data-end=\"2284\" \/>\u274c Trick: For all x, must find y so x*y = 1<\/p>\n<p data-start=\"2328\" data-end=\"2403\">Try x = 2 \u2192 y = 0.5 \u274c (Not integer)<br data-start=\"2363\" data-end=\"2366\" \/>Try x = 0 \u2192 y = undefined \u274c<br data-start=\"2393\" data-end=\"2396\" \/>\u274c FALSE<\/p>\n<hr data-start=\"2405\" data-end=\"2408\" \/>\n<h3 data-start=\"2410\" data-end=\"2443\">\ud83d\udd04 Summary Table for Shortcut<\/h3>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"2445\" data-end=\"2826\">\n<thead data-start=\"2445\" data-end=\"2476\">\n<tr data-start=\"2445\" data-end=\"2476\">\n<th data-start=\"2445\" data-end=\"2457\" data-col-size=\"sm\">Statement<\/th>\n<th data-start=\"2457\" data-end=\"2467\" data-col-size=\"sm\">Meaning<\/th>\n<th data-start=\"2467\" data-end=\"2476\" data-col-size=\"sm\">Trick<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2509\" data-end=\"2826\">\n<tr data-start=\"2509\" data-end=\"2602\">\n<td data-start=\"2509\" data-end=\"2527\" data-col-size=\"sm\"><code data-start=\"2511\" data-end=\"2526\">\u2200x \u2203y P(x, y)<\/code><\/td>\n<td data-col-size=\"sm\" data-start=\"2527\" data-end=\"2557\">For every x, there exists y<\/td>\n<td data-col-size=\"sm\" data-start=\"2557\" data-end=\"2602\">Try each x, find <strong data-start=\"2576\" data-end=\"2585\">any y<\/strong> that satisfies<\/td>\n<\/tr>\n<tr data-start=\"2603\" data-end=\"2697\">\n<td data-start=\"2603\" data-end=\"2621\" data-col-size=\"sm\"><code data-start=\"2605\" data-end=\"2620\">\u2203x \u2200y P(x, y)<\/code><\/td>\n<td data-col-size=\"sm\" data-start=\"2621\" data-end=\"2652\">Exists x such that for all y<\/td>\n<td data-col-size=\"sm\" data-start=\"2652\" data-end=\"2697\">Try some x, see if it works for <strong data-start=\"2686\" data-end=\"2695\">all y<\/strong><\/td>\n<\/tr>\n<tr data-start=\"2698\" data-end=\"2763\">\n<td data-start=\"2698\" data-end=\"2716\" data-col-size=\"sm\"><code data-start=\"2700\" data-end=\"2715\">\u2200x \u2200y P(x, y)<\/code><\/td>\n<td data-col-size=\"sm\" data-start=\"2716\" data-end=\"2740\">All combinations true<\/td>\n<td data-col-size=\"sm\" data-start=\"2740\" data-end=\"2763\"><strong data-start=\"2742\" data-end=\"2755\">Full grid<\/strong> check<\/td>\n<\/tr>\n<tr data-start=\"2764\" data-end=\"2826\">\n<td data-start=\"2764\" data-end=\"2782\" data-col-size=\"sm\"><code data-start=\"2766\" data-end=\"2781\">\u2203x \u2203y P(x, y)<\/code><\/td>\n<td data-col-size=\"sm\" data-start=\"2782\" data-end=\"2808\">One good pair is enough<\/td>\n<td data-col-size=\"sm\" data-start=\"2808\" data-end=\"2826\">Try few combos<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"2828\" data-end=\"2831\" \/>\n<h2 data-start=\"2833\" data-end=\"2853\">\ud83c\udfc1 <strong data-start=\"2839\" data-end=\"2853\">Conclusion<\/strong><\/h2>\n<p data-start=\"2855\" data-end=\"2903\">Nested quantifier problems become EASY when you:<\/p>\n<ol data-start=\"2905\" data-end=\"3066\">\n<li data-start=\"2905\" data-end=\"2928\">\n<p data-start=\"2908\" data-end=\"2928\">Start <strong data-start=\"2914\" data-end=\"2928\">inside out<\/strong><\/p>\n<\/li>\n<li data-start=\"2929\" data-end=\"2960\">\n<p data-start=\"2932\" data-end=\"2960\">Use <strong data-start=\"2936\" data-end=\"2952\">tiny domains<\/strong> to test<\/p>\n<\/li>\n<li data-start=\"2961\" data-end=\"3000\">\n<p data-start=\"2964\" data-end=\"3000\">Translate logic to <strong data-start=\"2983\" data-end=\"3000\">plain English<\/strong><\/p>\n<\/li>\n<li data-start=\"3001\" data-end=\"3034\">\n<p data-start=\"3004\" data-end=\"3034\">Use <strong data-start=\"3008\" data-end=\"3024\">truth tables<\/strong> if needed<\/p>\n<\/li>\n<li data-start=\"3035\" data-end=\"3066\">\n<p data-start=\"3038\" data-end=\"3066\">Remember \u2013 <strong data-start=\"3049\" data-end=\"3066\">order matters<\/strong><\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"3068\" data-end=\"3071\" \/>\n<p data-start=\"3073\" data-end=\"3202\" data-is-last-node=\"\" data-is-only-node=\"\">Would you like a <strong data-start=\"3090\" data-end=\"3105\">PDF version<\/strong>, <strong data-start=\"3107\" data-end=\"3125\">quiz worksheet<\/strong>, or <strong data-start=\"3130\" data-end=\"3151\">video explanation<\/strong> for practice? Let me know \u2013 I\u2019ll prepare that too!<\/p>\n<h3 data-start=\"3073\" data-end=\"3202\"><a href=\"https:\/\/www2.cs.uh.edu\/~arjun\/courses\/ds\/DiscMaths4CompSc.pdf\" target=\"_blank\" rel=\"noopener\">Day 03Part 04- Discrete mathematics &#8211; Short trick method for solving Nested Quantifiers problems.<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Day 03Part 04- Discrete mathematics &#8211; Short trick method for solving Nested Quantifiers problems. [fvplayer id=&#8221;161&#8243;] \u00a0Day 03 | Part 04 &#8211; Discrete Mathematics Short Trick Methods for Solving Nested Quantifiers Problems Nested quantifiers can be tricky, but with some simple techniques, you can solve them quickly and accurately. Let&#8217;s go through some short tricks [&hellip;]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-2892","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2892","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=2892"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2892\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2892"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2892"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2892"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}