Answer to Question #266701 in Discrete Mathematics for Alina

Question

Prove or disprove that if R and S are antisymmetric, then so is:

(a) (R ∪ S)

(b) (R ∩ S)

Answer 1

a)

If R and S are antisymmetric, then R ∪ S is antisymmetric. We disprove the statement.

(1) Let T = {a, b}, R = {(a, b)}, and S = {(b, a)}.

(2). R and S are antisymmetric.

(3). R ∪ S = {(a, b),(b, a)}.

(4). ∃a, b,(a, b) ∈ R ∪ S ∧ (b, a) ∈ R ∪ S ∧ a 6= b. (Step 3)

(5). R ∪ S is not antisymmetric.

b)

(R ∩ S) is antisymmetric

Let S contains element which is symmetric to some element in R. But this element will not be included in (R ∩ S) because S does not contain this element. So, R remains antsymmetric.

The same is for S.