Which of the following is (are) true for R given below? R= {(a,b)|both a and b are non-zero integers and a/b is an integer} a) R is a reflexive relation. b) R is a symmetric relation. c) R is a transitive relation. d) R is an equivalence relation.
Answers
Answer:
option b is correct please make me a brainlist
Step-by-step explanation:
The Relation R is a reflexive and transitive relation. The correct options are (a) R is a reflexive relation, and (c) R is a transitive relation.
Given:
R = {(a, b) | both a and b are non-zero integers and a/b is an integer}
To Find:
Whether the relation is reflexive, symmetric, transitive and equivalence relation.
Solution:
→ A relation R on a set A is called reflexive if (a, a) ∈ R for all a ∈ A.
→ A relation R is said to be symmetric if (a, b) ∈ R implies (b, a) ∈ R.
→ A relation R is said to be transitive if (a, b) and (b, c) belong to R implies that (a, c) also belongs to R.
→ If a relation R is reflexive, symmetric, as well as transitive, then it is known as an equivalence relation.
→ For our Relation:
R = {(a, b) | both a and b are non-zero integers and a/b is an integer}
→ (a, a) will belong to relation R because (a/a) is equal to 1, which is an integer. As (a, a) belongs to the relation R, therefore relation R will be a reflexive relation.
→ If (a, b)∈R, then this means a/b would be an integer. Hence b/a could not be an integer, therefore (b, a)∉R. So the relation R is not a symmetric relation.
→ Assume if (a,b) and (b,c) both belong to Relation R. This means a/b as well as b/c both will be an integer.
Since a/c is also an integer, therefore (a,c) would also belong to the relation R. Hence the relation R is a transitive relation.
Hence the Relation R is a reflexive and transitive relation. The correct options are (a) R is a reflexive relation, and (c) R is a transitive relation.
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