Consider the relation r = (3, 4) on the set S = (0, 1, 2, 3, 4) Which of the fol
ID: 3012940 • Letter: C
Question
Consider the relation r = (3, 4) on the set S = (0, 1, 2, 3, 4) Which of the following is true? r is reflexive and symmetric r is symmetric and antisymetric r is antisymetric and transitive r is transitive and reflexive Consider the binary relation rho on the set S = (2, 3, 6, 12, 18, 24) defined by m n if and only if m divides n. Which of the following is true? rho is an equivalence relation rho is a partial ordering rho is both, an equivalence relation and a partial ordering None of the above.Explanation / Answer
Solution 6:
The relation is in fact anti-symmetric, r = {(1,2)} on the set S = {0,1,2,3,4}.
Why is this anti-symmetric? Because in order for the relation to be anti-symmetric, it must be true that whenever some pair (x,y) with xy is an element of the relation R, then the opposite pair (y,x) cannot also be an element of R. This is true for our relation, since we have (3,4) S, but we don't have (4,3) in S.
Solution 7:
For instance that m Þ n if and only if m divides n, the relation Þ is both, an equivalence relation and a partial ordering.
As,
An equivalence relation on a set X is a binary relation that is reflexive, symmetric, and transitive.
A partition of X is a collection of subsets that are mutually exclusive and exhaustive.