Q. 7

# Let S be the set of all real numbers and letR = {(a, b) : a, b ∈ S and a = ± b}.Show that R is an equivalence relation on S.

In order to show R is an equivalence relation we need to show R is Reflexive, Symmetric and Transitive.

Given that, a, b S, R = {(a, b) : a = ± b }

Now,

R is Reflexive if (a,a) R a S

For any a S, we have

a = ±a

(a,a) R

Thus, R is reflexive.

R is Symmetric if (a,b) R (b,a) R a,b S

(a,b) R

a = ± b

b = ± a

(b,a) R

Thus, R is symmetric .

R is Transitive if (a,b) R and (b,c) R (a,c) R a,b,c S

Let (a,b) R and (b,c) R a, b,c S

a = ± b and b = ± c

a = ± c

(a, c) R

Thus, R is transitive.

Hence, R is an equivalence relation.

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