) Let set ? = {1,2,3,4} and let ?1 and ?2 be binary relations on ?. Specifically, let:
?1 = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 4), (3, 4), (4, 2), (4, 3) (4, 4)}
?2 = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (4, 1), (4, 2)}
Determine the following:
a) Whether ?1 is reflexive, irreflexive, symmetric, anti-symmetric and/or transitive.
b) Whether ?2 is reflexive, irreflexive, symmetric, anti-symmetric and/or transitive.
c) ?1 ∘ ?2.
d) ?2 ∘ ?1.
e) ?1 ∪ ?2.
f) ?1 ∩ ?2.
g) The reflexive, symmetric and transitive closures of both ?1 and ?2