Question

Suppose that R is a binary relation over the set A. For the following questions, fill in the blanks with one or more of the following words: reflexive transitive symmetric antisymmetric

a) If for all a ∈ A, (a, a) ∈ R, then R is
b) If for all a, b ∈ A, (a, b) ∈ R implies (b, a) ∈ R, then R is
c) R is if for all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then
(a, c) ∈ R.
d) R is if for all a, b ∈ R such that a 6= b, if (a, b) ∈ R then
(b, a) is not in R.
e) If R is an equivalence relation, then R is and and.
f) If R is a partial order, then R is and and .

Answer 1

In binary relations, if all elements in set A have a relation to themselves, then the relation is reflexive. If for any two elements a and b, if a has a relation to b then b has a relation to a, then the relation is symmetric.

If for any three elements a, b, and c, if a has a relation to b and b has a relation to c, then a has a relation to c, then the relation is transitive.

Step-by-step explanation:

a) If for all a ∈ A, (a, a) ∈ R, then R is reflexive.

b) If for all a, b ∈ A, (a, b) ∈ R implies (b, a) ∈ R, then R is symmetric.

c) R is transitive if for all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R.

d) R is antisymmetric if for all a, b ∈ R such that a ≠ b, if (a, b) ∈ R then (b, a) is not in R.

e) If R is an equivalence relation, then R is reflexive, symmetric, and transitive.

f) If R is a partial order, then R is reflexive, antisymmetric, and transitive.