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Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 16 Maths Textbook Solution

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Answer:R\cup S\: is\: not\: transitive
Hint:  For transitive relation
x=y \: and\: y=z, then\: also\: x=z.
Given:\\\text{R and S are transitive relation on a set A}
To Prove:R\cup S\: \text{ may not be a transitive relation on A}
We will prove this by means of an example.
Let \: A=\left \{ a, b, c \right \} be\: a\: set
R= \left \{ (a,a),(b, b),(c, c),(a,b),(b, a)\right \} and
S=\left \{ (a, a),(b, b),(c,c),(b, c),(c, b) \right \} are\: two\: relations\: on\: A
\\\text{Clearly, R and S are transitive relation on A.}       
Now, R\cup S=\left \{ (a, a),(b, b),(c, c),(a, b),(b, a),(b, c),(c, b) \right \}
Here, (a, b) \: \epsilon \: R\cup S\: and \: (b, c)\: \epsilon\: R\cup S
but \: (a,c) \: \notin\: R\cup\: S
R\cup S\: is\: not\: transitive


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