{"id":3109,"date":"2025-06-09T10:15:35","date_gmt":"2025-06-09T10:15:35","guid":{"rendered":"https:\/\/diznr.com\/?p=3109"},"modified":"2025-06-09T10:15:35","modified_gmt":"2025-06-09T10:15:35","slug":"day-02-discrete-mathematics-for-computer-science-gate-ir-reflexive-relation-with-understanding-basic","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-02-discrete-mathematics-for-computer-science-gate-ir-reflexive-relation-with-understanding-basic\/","title":{"rendered":"Day 02- Discrete mathematics for computer science gate-Ir-reflexive Relation with basic understanding"},"content":{"rendered":"

Day 02- Discrete mathematics for computer science gate-Ir-reflexive Relation with basic understanding<\/p>\n

[fvplayer id=”257″]<\/p>\n

Day 02: Discrete Mathematics for Computer Science (GATE) \u2013 Irreflexive Relation with Basic Understanding<\/strong><\/h3>\n

\u00a0What is a Relation in Discrete Mathematics?<\/strong><\/h4>\n

In Discrete Mathematics, a relation<\/strong> is a set of ordered pairs that define a connection between elements of two sets. If a relation is defined on a single set, it is called a binary relation<\/strong>.<\/p>\n

\u00a0What is an Irreflexive Relation?<\/strong><\/h3>\n

A relation R on a set A<\/strong> is called irreflexive<\/strong> if no element<\/strong> in the set is related to itself. That is, for every element aa<\/span>a<\/span><\/span><\/span><\/span> in AA<\/span>A<\/span><\/span><\/span><\/span>:<\/p>\n

(a,a)\u2209R(a, a) \\notin R<\/span>(<\/span>a<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span>\/<\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span><\/span><\/p>\n

This means that self-loops are not allowed<\/strong> in an irreflexive relation.<\/p>\n

\u00a0Example:<\/strong><\/h4>\n

Let A = {1, 2, 3}<\/strong> and let R<\/strong> be a relation on A:<\/p>\n

R={(1,2),(2,3),(3,1)}R = \\{(1,2), (2,3), (3,1)\\}<\/span>R<\/span>=<\/span><\/span>{(<\/span>1<\/span>,<\/span>2<\/span>)<\/span>,<\/span>(<\/span>2<\/span>,<\/span>3<\/span>)<\/span>,<\/span>(<\/span>3<\/span>,<\/span>1<\/span>)}<\/span><\/span><\/span><\/span><\/span><\/p>\n

Since (1,1), (2,2), and (3,3) are NOT present in R<\/strong>, the relation is irreflexive<\/strong>.<\/p>\n

\u00a0Difference Between Reflexive and Irreflexive Relations<\/strong><\/h3>\n\n\n\n\n\n\n\n
Property<\/th>\nReflexive Relation<\/th>\nIrreflexive Relation<\/th>\n<\/tr>\n<\/thead>\n
Definition<\/td>\nEvery element relates to itself<\/td>\nNo element relates to itself<\/td>\n<\/tr>\n
Condition<\/td>\n(a,a)\u2208R(a, a) \\in R<\/span>(<\/span>a<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span> for all a\u2208Aa \\in A<\/span>a<\/span>\u2208<\/span><\/span>A<\/span><\/span><\/span><\/span><\/td>\n(a,a)\u2209R(a, a) \\notin R<\/span>(<\/span>a<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span>\/<\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span> for all a\u2208Aa \\in A<\/span>a<\/span>\u2208<\/span><\/span>A<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
Example<\/td>\nR={(1,1),(2,2),(3,3),(1,2)}R = \\{(1,1), (2,2), (3,3), (1,2)\\}<\/span>R<\/span>=<\/span><\/span>{(<\/span>1<\/span>,<\/span>1<\/span>)<\/span>,<\/span>(<\/span>2<\/span>,<\/span>2<\/span>)<\/span>,<\/span>(<\/span>3<\/span>,<\/span>3<\/span>)<\/span>,<\/span>(<\/span>1<\/span>,<\/span>2<\/span>)}<\/span><\/span><\/span><\/span><\/td>\nR={(1,2),(2,3),(3,1)}R = \\{(1,2), (2,3), (3,1)\\}<\/span>R<\/span>=<\/span><\/span>{(<\/span>1<\/span>,<\/span>2<\/span>)<\/span>,<\/span>(<\/span>2<\/span>,<\/span>3<\/span>)<\/span>,<\/span>(<\/span>3<\/span>,<\/span>1<\/span>)}<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u00a0Special Cases:<\/strong><\/h3>\n

A relation can be neither reflexive nor irreflexive<\/strong> if some but not all elements relate to themselves.
An empty relation<\/strong> is always irreflexive because it contains no pairs at all.<\/p>\n

\u00a0Importance of Irreflexive Relations in Computer Science<\/strong><\/h3>\n

\u00a0Used in graph theory<\/strong> for modeling directed graphs<\/strong> without self-loops.
\u00a0Important in order relations<\/strong> like strict partial order<\/strong> and strict preference relations<\/strong>.
\u00a0Helps in programming languages & automata theory<\/strong> for defining transitions without self-loops<\/strong>.<\/p>\n

Would you like practice problems or more examples<\/strong>?<\/p>\n

Day 02- Discrete mathematics for computer science gate-Ir-reflexive Relation with basic understanding<\/a><\/h3>\n

Notes on Discrete Mathematics<\/a><\/h3>\n

Here is a clear and beginner-friendly explanation of Day 02 \u2013 Discrete Mathematics for Computer Science (GATE)<\/strong>, focused on Irreflexive Relation<\/strong>, with basic understanding.<\/p>\n

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\ud83d\udcd8 Discrete Mathematics \u2013 Day 02 Topic: Irreflexive Relations<\/strong><\/h2>\n
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\ud83d\udd0d What is a Relation? (Quick Recap)<\/strong><\/h3>\n

In set theory, a relation<\/strong> is a subset of a Cartesian product<\/strong> of a set with itself.<\/p>\n

For a set A<\/strong>, a relation R<\/strong> is a subset of A\u00d7AA \\times A<\/span>, meaning:<\/p>\n

R\u2286A\u00d7AR \\subseteq A \\times A<\/span><\/p>\n

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\ud83d\udd01 Reflexive vs Irreflexive Relation<\/strong><\/h2>\n

\u2705 Reflexive Relation:<\/strong><\/h3>\n

A relation RR<\/span> on set AA<\/span> is reflexive<\/strong> if:<\/p>\n

\u2200a\u2208A,\u00a0(a,a)\u2208R\\forall a \\in A,\\ (a, a) \\in R<\/span><\/p>\n

That means every element is related to itself<\/strong>.<\/p><\/blockquote>\n

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\ud83d\udeab Irreflexive Relation:<\/strong><\/h3>\n

A relation RR<\/span> on set AA<\/span> is irreflexive<\/strong> if:<\/p>\n

\u2200a\u2208A,\u00a0(a,a)\u2209R\\forall a \\in A,\\ (a, a) \\notin R<\/span><\/p>\n

That means no element is related to itself<\/strong>.<\/p><\/blockquote>\n

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\ud83d\udd01\ud83c\udd9a\ud83d\udeab Summary:<\/h3>\n\n\n\n\n\n\n
Property<\/th>\nCondition<\/th>\nExample<\/th>\n<\/tr>\n<\/thead>\n
Reflexive<\/td>\nEvery element maps to itself<\/td>\n(a, a) \u2208 R for all a \u2208 A<\/td>\n<\/tr>\n
Irreflexive<\/td>\nNo element maps to itself<\/td>\n(a, a) \u2209 R for all a \u2208 A<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
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\ud83d\udcda Example of Irreflexive Relation<\/strong><\/h2>\n

Let A={1,2,3}A = \\{1, 2, 3\\}<\/span><\/p>\n

Let R={(1,2),(2,3),(3,1)}R = \\{(1, 2), (2, 3), (3, 1)\\}<\/span><\/p>\n

Now, check:<\/p>\n