On A = {1, 2, 3, 4}, let R = {(a,b): a + b is even}. What type of relation is R?
Answer and explanation
Correct answer: Transitive
The key concept is parity. The sum a+b is even exactly when a and b have the same parity: both are even or both are odd. Suppose (a,b) ∈ R and (b,c) ∈ R. Then a and b have the same parity, and b and c also have the same parity. Consequently, a and c have the same parity, so a+c is even and (a,c) ∈ R. This proves transitivity, making option A correct. In fact, the relation is also reflexive and symmetric, but those facts are not needed for this question.
Frequently asked questions
What is the correct answer to this question?
Transitive
Why is this the correct answer?
The key concept is parity. The sum a+b is even exactly when a and b have the same parity: both are even or both are odd. Suppose (a,b) ∈ R and (b,c) ∈ R. Then a and b have the same parity, and b and c also have the same parity. Consequently, a and c have the same parity, so a+c is even and (a,c) ∈ R. This proves transitivity, making option A correct. In fact, the relation is also reflexive and symmetric, but those facts are not needed for this question.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Transitive relation.
