Let A={1,2,3} and a relation

Solution

Given the set A={1,2,3} and the relation R={(1,1),(2,2),(3,3),(1,2),(2,1)} on A. We need to determine which of the given properties is false for R. (A)Reflexive: A relation R on a set A is reflexive if for every element a ∈ A, (a,a) ∈ R. For A={1,2,3}, the relation R must contain (1,1),(2,2),(3,3) to be reflexive. R contains (1,1),(2,2),(3,3). Therefore, R is reflexive. So, (A) is true. (B)Symmetric: A relation R on a set A is symmetric if for every (a,b) ∈ R, it implies that (b,a) ∈ R. In R:

• (1,2) ∈ R, and (2,1) ∈ R.
• (1,1) ∈ R, and (1,1) ∈ R.
• (2,2) ∈ R, and (2,2) ∈ R.
• (3,3) ∈ R, and (3,3) ∈ R. All pairs in R satisfy the condition for symmetry. Therefore, R is symmetric. So, (B) is true.

(C)Transitive: A relation R on a set A is transitive if for every (a,b) ∈ R and (b,c) ∈ R, it implies that (a,c) ∈ R. We need to check all possible pairs (a,b) and (b,c) in R:

• (1,2) ∈ R and (2,1) ∈ R. For transitivity, (1,1) should be in R, which it is.
• (2,1) ∈ R and (1,2) ∈ R. For transitivity, (2,2) should be in R, which it is.
• (1,2) ∈ R and (2,2) ∈ R. For transitivity, (1,2) should be in R, which it is.
• (2,1) ∈ R and (1,1) ∈ R. For transitivity, (2,1) should be in R, which it is.
• (1,1) ∈ R and (1,2) ∈ R. For transitivity, (1,2) should be in R, which it is.
• (2,2) ∈ R and (2,1) ∈ R. For transitivity, (2,1) should be in R, which it is.
• (1,1) ∈ R and (1,1) ∈ R. For transitivity, (1,1) should be in R, which it is.
• (2,2) ∈ R and (2,2) ∈ R. For transitivity, (2,2) should be in R, which it is.
• (3,3) ∈ R and (3,3) ∈ R. For transitivity, (3,3) should be in R, which it is. All cases satisfy the condition for transitivity. Therefore, R is transitive. So, (C) is true.

(D)Antisymmetric: A relation R on a set A is antisymmetric if for every (a,b) ∈ R and (b,a) ∈ R, it implies that a=b. In R, we have (1,2) ∈ R and (2,1) ∈ R. For R to be antisymmetric, it must be that 1=2, which is false. Therefore, R is not antisymmetric. So, (D) is false.