Question

Show that the relation in the set R of real numbers, defined as
R = {(a,b): a Sb) is neither reflexive nor symmetric nor transitive.​

Answer 1

Answer:

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Step-by-step explanation:

A relation is said to be reflexive relation, if every element of A is related to itself.

thus,(a,a) ∈ R , for all a∈ A => R is reflexive

in given relation if we take (1/2,1/2)

we observed , 1/2 1/2²

hence, it is not relfexive relation.

A relation is said to be symmetric relation, if

(a, b) ∈ R => (b, a) ∈ R , for all a, b ∈ A

e.g., aRb => bRa, for all a, b ∈ A => R is symmetric.

if we take (a, b) = (1,4)

then, (b , a) = (4, 1) doesn't belong to relation

because 4² 1²

A relation is said to be transitive relation if

(a, b) ∈ R and (b, c) ∈ R => (a,c) ∈ R for all a,b,c ∈ A

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