Suppose we have binary relation on natural number set N×N
Suppose we have a binary relation R on a natural number set N×N which is defined as follows:
(x,y)R(z,w) if x≤z or y≤w
Let us consider the following propositions:
A: R is reflexive.
B: R is symmetric.
C: R is transitive.
- A.
Both A and B are true
- B.
A is true and C is false
- C.
A is false and C is true
- D.
Both A and B are false
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Correct answer: B
Conclusion: The relation is reflexive, not symmetric, and not transitive; therefore reflexivity is true and transitivity is false.
Reflexive: For any (x,y) in N×N, x ≤ x, so (x,y)R(x,y) holds.
Not symmetric: Example: (1,1)R(2,2) because 1 ≤ 2, but (2,2)R(1,1) is false because neither 2 ≤ 1 nor 2 ≤ 1 holds.
Not transitive: Example: (3,3)R(3,0) since 3 ≤ 3, and (3,0)R(2,1) since 0 ≤ 1, but (3,3)R(2,1) is false because 3 ≤ 2 and 3 ≤ 1 are both false.
Therefore: A (reflexive) is true; B (symmetric) is false; C (transitive) is false.