The number of relations on the set $A=\{1,2,3\}$, containing at most 6 elements including $(1,2)$, which are reflexive and transitive but not symmetric, is __________.
Answer (integer)
6
Solution
<p>Since relation needs to be reflexive the ordered pairs $(1,1),(2,2),(3,3)$ need to be there and $(1,2)$ is also to be included.</p>
<p>Let's call $R_0=\{(1,1),(2,2),(3,3),(1,2)\}$ the base relation.</p>
<p>$\because A \times A$ contain $3 \times 3=9$ ordered pairs, remaining 5 ordered are</p>
<p>$2,1),(1,3),(3,1),(2,3),(3,2)$</p>
<p>We have to add at most two ordered pairs to $R_0$ such that resulting relation is reflexive, transitive but not symmetric.</p>
<p>Following are the only possibilities.</p>
<p>$R=R_0 U\{(1,3)\}$</p>
<p>OR $R_0 U\{(3,2)\}$</p>
<p>OR $R_0 U\{(1,3),(3,1)\}$</p>
<p>OR $R_0 U\{(1,3),(3,2)\}$</p>
<p>OR $R_0 U\{(3,1),(3,2)\}$</p>
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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