Sure! Let’s explore First Order Predicate Logic (FOPL)<\/strong>, also known as Predicate Calculus<\/strong>, with clear explanations and examples \u2014 as typically introduced on Day 5 of a Discrete Mathematics course<\/strong>.<\/p>\n Predicate Logic is an extension of Propositional Logic<\/strong>. Universal Quantifier (\u2200):<\/strong> Existential Quantifier (\u2203):<\/strong> Let:<\/p>\n Domain: All humans<\/p>\n<\/li>\n Predicate: Statement: “All humans are mortal” Let:<\/p>\n Domain: All integers<\/p>\n<\/li>\n Predicate: Statement: “There exists an even number” Statement: “All numbers are even” Let:<\/p>\n Domain: All people<\/p>\n<\/li>\n Predicate:
\n\ud83d\udcd8 Day 05 Part 01 \u2013 First Order Predicate Logic (FOPL)<\/strong><\/h2>\n
\ud83d\udd39 What is Predicate Logic?<\/h3>\n
Unlike propositional logic, which deals with simple, atomic statements (e.g., \u201cIt is raining\u201d), predicate logic<\/strong> deals with statements that include variables and quantifiers<\/strong>.<\/p>\n
\n\u2705 Basic Terms:<\/strong><\/h3>\n
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\n \nTerm<\/th>\n Meaning<\/th>\n<\/tr>\n<\/thead>\n \n Predicate<\/strong><\/td>\n A property or relation expressed about a subject (e.g., P(x): “x is tall”)<\/td>\n<\/tr>\n \n Variable<\/strong><\/td>\n A symbol like x, y, z representing an element from a domain<\/td>\n<\/tr>\n \n Quantifier<\/strong><\/td>\n Symbol indicating “for all” (\u2200) or “there exists” (\u2203)<\/td>\n<\/tr>\n \n Domain<\/strong><\/td>\n The set of all possible values a variable can take<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n\ud83d\udd39 Two Quantifiers in Predicate Logic<\/strong><\/h3>\n
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Means “for all”<\/strong> or “every”<\/strong>
Symbol: \u2200x P(x)<\/code> \u2192 P(x) is true for all values of x<\/em><\/p>\n<\/li>\n
Means “there exists”<\/strong>
Symbol: \u2203x P(x)<\/code> \u2192 There exists at least one x for which P(x) is true<\/em><\/p>\n<\/li>\n<\/ol>\n
\n\ud83d\udd0e Example 1 \u2013 Simple Predicate<\/strong><\/h3>\n
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M(x): \"x is mortal\"<\/code>
Then:<\/p>\n<\/li>\n
\u2192 Symbolic form: \u2200x M(x)<\/code><\/p>\n<\/li>\n<\/ul>\n
\n\ud83d\udd0e Example 2 \u2013 Using \u2203 and \u2200<\/strong><\/h3>\n
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E(x): \"x is even\"<\/code>
Then:<\/p>\n<\/li>\n
\u2192 Symbolic form: \u2203x E(x)<\/code><\/p>\n<\/li>\n
\u2192 Symbolic form: \u2200x E(x)<\/code> (This is false<\/strong>!)<\/p>\n<\/li>\n<\/ul>\n
\n\ud83d\udd0e Example 3 \u2013 With Two Variables<\/strong><\/h3>\n
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