A relation R is defined on the set of integers ’ Z ‘ by “a R b if a – b is divisible by 5” for a, b ∈ Z. Examine if R is an equivalence relation on Z or not choice
Answers
Answer:Equivalence relation on set is a relation which is reflexive, symmetric and transitive.
A relation R, defined in a set A, is said to be an equivalence relation if and only if
(i) R is reflexive, that is, aRa for all a ∈ A.
(ii) R is symmetric, that is, aRb ⇒ bRa for all a, b ∈ A.
(iii) R is transitive, that is aRb and bRc ⇒ aRc for all a, b, c ∈ A.
The relation defined by “x is equal to y” in the set A of real numbers is an equivalence relation.
Let A be a set of triangles in a plane. The relation R is defined as “x is similar to y, x, y ∈ A”.
We see that R is;
(i) Reflexive, for, every triangle is similar to itself.
(ii) Symmetric, for, if x be similar to y, then y is also similar to x.
(iii) Transitive, for, if x be similar to y and y be similar to z, then x is also similar to z.
Hence R is an equivalence relation.
A relation R in a set S is called a partial order relation if it satisfies the following conditions:
(i) aRa for all a∈ A, [Reflexivity]
(ii) aRb and bRa ⇒ a = b, [Anti-symmetry]
(iii) aRb and bRc ⇒ aRc, [Transitivity]
In the set of natural numbers, the relation R defined by “aRb if a divides b” is a partial order relation, since here R is reflexive, anti-symmetric and transitive.
A set, in which a partial order relation is defined, is called a partially ordered set or a poset.
Step-by-step explanation: