Consider the set A = {1, 2, 3, 4}. For each of the properties below, ive an exam
ID: 2967287 • Letter: C
Question
Consider the set A = {1, 2, 3, 4}. For each of the properties below, ive an example of a relation on A having that property. In each case, represent the relation by drawing a grid. (a) A relation which is symmetric and reflexive, but not transitive. (Explain why it is not transitive.) (b) A relation which is neither symmetric nor anti-symmetric. (Explain why it is neither symmetric nor anti-symmetric.) (c) A relation which is transitive but not reflexive, and such that the relation (thought of as a subset of A x A) has at least five elements. (Explain why it is not reflexive.)Explanation / Answer
(a) R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2)}
The first 4 ordered pairs are required to ensure reflexivity. The relation is not transitive since (1, 2) and (2, 3) are in R but (1, 3) is not in R. The relation is symmetric because any time we have (a, b) in the relation, we also have (b, a) in the relation.
(b) R = {(1, 2), (2, 1), (3, 4)}
R is not symmetric because (3, 4) is in R but (4, 3) is not. R is not antisymmetric because there exist a pair of elements of A (namely 1 and 2) such that both (1, 2) and (2, 1) are in R.
(c) R = {(1, 2), (2, 3), (1, 3), (1, 1), (2, 2), (4, 4)}
This relation is not reflexive because there exists an element of A (namely 3) that is not related to itself, ie (3, 3) is not in R.