GATE CS 2016 Set 2 — Question 36
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Engineering Mathematics → Sets & Combinatorics → Sets, Relations & Functions
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Question
A binary relation on is defined as follows: if or . Consider the following propositions:P: is reflexive
Q: is transitiveWhich one of the following statements is TRUE?
Q: is transitiveWhich one of the following statements is TRUE?
Correct answer
(B) P is true and Q is false.
Solution
We are given the relation on defined by .Reflexivity (P):
For to be reflexive, must be true for all .
.
Since is always true, the disjunction is true. Thus, is reflexive. P is true.Transitivity (Q):
For to be transitive, if and , then .
Let's look for a counterexample.
Let .
Let .
Let .
For to be reflexive, must be true for all .
.
Since is always true, the disjunction is true. Thus, is reflexive. P is true.Transitivity (Q):
For to be transitive, if and , then .
Let's look for a counterexample.
Let .
Let .
Let .
1.Check : . Since is true, this holds.
2.Check : . Since is true, this holds.
3.Check : . Both are false, so this does not hold.
Since we found a counterexample, is not transitive. Q is false.Therefore, P is true and Q is false.Continue learning with Success Tracker
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