Question

Determine whether the relation R on N is reflexive, symmetric, and/or transitive. Prove your answer.

a)R = x,2.
b)R = {(x,y) : x,y ∈ A}. A = {1,2,3,4}
c)R = {(x,y) : x,y ∈N,x is even ,y is odd }.

Answer 1

The relations in question a) and b) are reflexive, with relation b) also being symmetric and transitive. Relation c) is neither reflexive, symmetric, nor transitive.

Step-by-step explanation:

The student's question involves determining whether the given relations are reflexive, symmetric, and/or transitive. Let us examine each relation one by one.

This relation is reflexive because every even number is related to itself. It's also symmetric because if (x, y) is in the relation, then (y, x) is also in the relation since both x and y are even. However, it's not necessarily transitive because there might be numbers that are not related. For example, if a, b, c are 4, 4, and 6 respectively, we have (4,4) and (4,6) in R, but not (4,6).

This relation is reflexive, symmetric, and transitive because it's the full set of ordered pairs from A, essentially the relation of equivalence in the set A.

This relation is not reflexive because no number is both even and odd. It's not symmetric because if (x, y) is in R, (y, x) can't be in R since y is odd and x is even. It's also not transitive because if (a, b) and (b, c) are in R, a and c can't be in relation since there are no b that are both even and odd.