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A binary relation R on N × N is defined as follows: (a,b)R(c,d) if a ≤ c or b ≤ d. Consider the following propositions:
P: R is reflexive
Q: R is transitive
Which one of the following statements is TRUE?
- Both P and Q are true.
- P is true and Q is false.
- P is false and Q is true.
- Both P and Q are false.
Correct answer: P is true and Q is false.
Solution
The relation R is reflexive because for any pair (a,b), it holds that (a,b)R(a,b) since a ≤ a and b ≤ b. However, R is not transitive because if (a,b)R(c,d) and (c,d)R(e,f), it does not necessarily follow that (a,b)R(e,f) since the conditions can fail when combining the inequalities.
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