Question

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?
1. Both P and Q are true.
2. P is true and Q is false.
3. P is false and Q is true.
4. Both P and Q are false.

Answer 1

Correct Answer - Option 2 : P is true and Q is false.

Concept:

Reflexive relation: Reflexive relation on a set is a binary element in which every element is related to itself. Let A be a set and R be the relation defined in it. R is set to be reflexive, if (a,a) ϵ R for all a ϵ A i.e. every element of A is related to itself.

Transitive relation: A binary relation R on a set A is transitive if whenever an element ‘a’ is related to an element ‘b’ and ‘b’ in turn is related to an element ‘c’, then a is also related to ‘c’.

For all, a, b, c ϵ A if aRb and bRc then aRc.

Explanation:

Here, it is given that:

(a, b)R(c, d) if a ≤ c or b ≤ d

Statement P: R is reflexive → TRUE

For R to be reflexive (a, b) R (a, b) should hold true.

Here, a ≤ a or b ≤ b

So, (a, b) R (a, b) holds true. R is reflexive.

Statement Q: R is transitive → FALSE

Consider the elements as (4, 5), (5, 1), (1, 1)

As, (4, 5) R (5, 1) so, 4 ≤ 5 or 5 ≤ 1

And (5, 1) R (1, 1) so, 1 ≤ 1

But here, (4, 5) R (1, 1) where, 4 ≥ 1 and 5 ≥ 1

For this, relation doesn’t hold true,

So, Relation R is not transitive.