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If R is a relation on the set A = {1, 2, 3} defined by R = {(1, 2), (2, 3), (3,

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Question: If R is a relation on the set A = {1, 2, 3} defined by R = {(1, 2), (2, 3), (3, 1)}, which of the following properties does R possess?

Options:

  1. Reflexive
  2. Symmetric
  3. Transitive
  4. All of the above

Correct Answer: Transitive

Solution: 

R is not reflexive as (1,1), (2,2), (3,3) are not in R. It is symmetric as (2,3) implies (3,2) is not in R. It is transitive as (1,2) and (2,3) implies (1,3) is not in R. Thus, R is not all of the above.

If R is a relation on the set A = {1, 2, 3} defined by R = {(1, 2), (2, 3), (3,

Practice Questions

Q1
If R is a relation on the set A = {1, 2, 3} defined by R = {(1, 2), (2, 3), (3, 1)}, which of the following properties does R possess?
  1. Reflexive
  2. Symmetric
  3. Transitive
  4. All of the above

Questions & Step-by-Step Solutions

If R is a relation on the set A = {1, 2, 3} defined by R = {(1, 2), (2, 3), (3, 1)}, which of the following properties does R possess?
  • Step 1: Identify the set A, which is {1, 2, 3}.
  • Step 2: Identify the relation R, which is {(1, 2), (2, 3), (3, 1)}.
  • Step 3: Check if R is reflexive. A relation is reflexive if every element in A relates to itself. We need (1,1), (2,2), and (3,3) to be in R. They are not, so R is not reflexive.
  • Step 4: Check if R is symmetric. A relation is symmetric if for every (a, b) in R, (b, a) is also in R. We have (2, 3) in R, but (3, 2) is not in R, so R is not symmetric.
  • Step 5: Check if R is transitive. A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R. We have (1, 2) and (2, 3) in R, but (1, 3) is not in R, so R is not transitive.
  • Step 6: Conclude that R is not reflexive, not symmetric, and not transitive.
  • Reflexivity – A relation R on a set A is reflexive if every element in A is related to itself, meaning (a, a) is in R for all a in A.
  • Symmetry – A relation R is symmetric if for every (a, b) in R, the pair (b, a) is also in R.
  • Transitivity – A relation R is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R.
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