Example 1 · level 1
Which statement describes the relation R = {(a, b) : |a − b| ≤ 3} on the set of integers Z?
- Reflexive and symmetric, but not transitive
- Reflexive, but not symmetric or transitive
- Neither reflexive, symmetric nor transitive
- Transitive, but not reflexive or symmetric
Show solution
- |a − a| = 0 ≤ 3: reflexive.
- |a − b| = |b − a|: symmetric.
- (0, 3) and (3, 6) are in R but |0 − 6| = 6 > 3: not transitive.
- So R is: reflexive and symmetric, but not transitive.
Answer: Reflexive and symmetric, but not transitive
Example 2 · level 2
Let A = {1, 2, 3} and R = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} be a relation on A. Which statement is true?
- Reflexive and symmetric, but not transitive
- Neither reflexive, symmetric nor transitive
- Reflexive and transitive, but not symmetric
- An equivalence relation (reflexive, symmetric and transitive)
Show solution
- Every (a, a) for a in A is in R: reflexive.
- Every pair has its reverse in R: symmetric.
- Whenever (a, b) and (b, c) are in R, (a, c) is also in R: transitive.
- So R is: an equivalence relation (reflexive, symmetric and transitive).
Answer: An equivalence relation (reflexive, symmetric and transitive)
Example 3 · level 3
Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 1), (2, 2)} be a relation on A. Which statement is true?
- Reflexive and symmetric, but not transitive
- Reflexive and transitive, but not symmetric
- Reflexive, but not symmetric or transitive
- Symmetric, but not reflexive or transitive
Show solution
- (1, 1) is not in R: not reflexive.
- Every pair has its reverse in R: symmetric.
- (1, 2) and (2, 1) are in R but (1, 1) is not: not transitive.
- So R is: symmetric, but not reflexive or transitive.
Answer: Symmetric, but not reflexive or transitive