Help with Discrete math!

Question

A = Cartesian product Z x Z

F = a relation defined on A as follows:

For all (x1, y1) and (x2, y2) ∈ A : (x1, y1) F (x2, y2) ß> x1 = x2

Determine whether or not the given relation is:

(a) Reflexive:

(b) Symmetric:

(c) Anti-symmetric:

(d) Transitive:

Answer 1

Assuming the relation F is

(x1, y1) F (x2, y2) if and only if x1 = x2,

Is this relation reflexive?

Yes, as (x1, y1) F (x1, y1) because x1 = x1.

Is this relation symmetric?

Yes, because if (x1, y1) F (x2, y2), then (x2, y2) F (x1, y1).

This follows from the fact that if x1 = x2, then x2 = x1.

Is this relation antisymmetric?

No, because for F to be antisymmetric we would need

(x1, y1) F (x2, y2) and (x2, y2) F (x1, y1) implies (x1, y1) = (x2, y2). But this is false.

For example (1, 10) F (1, 20) and (1, 20) F (1, 10), but (1, 10) not= (1, 20).

Is this relation transitive?

Yes, because (x1,y1) F (x2, y2) and (x2, y2) F (x3, y3) does imply (x1, y1) F (x3, y3).

This follows because x1 = x2 and x2 = x3 implies x1 = x3.