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Let A = { a,b,c,d } which of the following is not true ?
a) R1 = { (a,a) (c,c) } is Symmetric, Anti-Symmetric and Transitive on A
b) R2 = { (a,b) (b,a) (a,c) (c,a) (c,d) } is Symmetric and Anti-Symmetric
c) R3 = { (b,c) (c,b) (d,d) } is Symmetric but not Anti-Symmetric
d) R4 = { (a,b) (b,c) (c,c) } is Anti-Symmetric but not Symmetric

2 Answers

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R2 is false.

Given relation R2 is Symmetric but not Anti-Symmetric. For a relation to be anti-symmetric, for any $(a, b) \in R$ and $(b, a) \in R$ implies that $a = b$, but here $(a, b) \in R$ and $(b, a) \in R$, but $a \neq b$
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Answer is (b)

 

SYMMETRIC:- Symmetric can contains both pair of elements as well as diagonal elements. e.g. (1,1) (1,2) (2,1)

ANTI-SYMMETRIC :- It contains only one set of elements from both pairs and diagonals elements. e.g. (1,1) (1,2) (2,3)

TRANSITIVE :- If the set contains elements like (1,2) and (2,3) then it should contain (1,3) as well. If not then it is fine. It accepts diagonal elements as well as other elements. e.g. (1,2) (2,3) (1,3) (1,1) (2,4)

 

a) R1 = { (a,a) (c,c) } is Symmetric, Anti-Symmetric and Transitive on A

This is True because this satisfies the property of all the above relations.

b) R2 = { (a,b) (b,a) (a,c) (c,a) (c,d) } is Symmetric and Anti-Symmetric

This is false coz (a,b) (b,a) are present which violates the property of Anti-Symmetric and (c,a) (c,d) are present for which (a,c) (d,c) are not present.

c) R3 = { (b,c) (c,b) (d,d) } is Symmetric but not Anti-Symmetric

This is true.

d) R4 = { (a,b) (b,c) (c,c) } is Anti-Symmetric but not Symmetric

This is true coz the order pairs sets of (a,b) is (b,a) and (b,c) is (c,b) which is not present, hence not satisfying the property of Symmetric.

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