Consider the relation R on the set S = {1, 2, 3, 4} defined by

R = { (1,2), (1,3), (1, 4), (2, 1), (3,1), (3, 4), (4,1), (4,2) }.

a. Explain why R is or is not symmetric.

b. Explain why R is or is not antisymmetric.

c. Explain why R is or is not reflexive.

Step-by-step explanation:

a) A relation R is symmetric when it includes the inverse relation, for example if it includes (8,9) then it should also include (9,8), if not, then the relation is not symmetric, you can see that in this case the relation includes (3,4) but not (4,3), therefore it is not symmetric

b) A relation is antisymmetric when it never includes the inverse relation, for example if it includes (8,9) then it can not include (9,8), if it does then it is not antisymmetric. In this case you can see that it first starts with (1,2) but then it also includes (2,1) so then it is not antisymmetric

c) A relation is reflexive if for each number of the domain set it includes the pair that is two times that same number, for example if 8 is in the domain then the relation should include (8,8). if not then it is not reflexive. In this case you can see that the domain S includes 1 but (1,1) is never on the relation or for example (2,2) is also never in the relation.