Q. 24.1( 159 Votes )
Show that the relation R in the set R of real numbers, defined as
R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.
Answer :
It is given that R = {(a, b) : a ≤ b2}
Check for reflexive:
We can see that 
Since, 
Therefore, R is not reflexive.
Check for symmetric:Now, (1,4) ϵ R as 1 < 42
But 4 is not less than 12.
Then, (4,1) ∉ R
Therefore, R is not symmetric.
Check for transitive:Now, (3, 2), (2, 1.5) ϵ R
But, 3 > (1.5)2 = 2.25.
Then, (3, 1.5) ∉ R
Therefore, R is not transitive.
Therefore, R is neither reflexive, nor symmetric, nor transitive.
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