Let S be the set of all real numbers and letR = {(a, b): a, b ∈ S an

1.	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.
Answer» We know that a = ± b and a² = b² are equal It is given that {(a, b): a, b ∈ S and a² = b²} Here, (a, a) ∈ R and a² = a² is true. Hence, R is reflexive where (a, b) ∈ R We know that a² = b² and b² = a² where (b, a) ∈ R Hence, R is symmetric. If (a, b) ∈ R and (b, c) ∈ R We know that a² = b² and b² = c² we get a² = c² where (a, c) ∈ R Hence, R is transitive.

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.

Answer»

We know that a = ± b and a² = b² are equal

It is given that {(a, b): a, b ∈ S and a² = b²}

Here, (a, a) ∈ R and a² = a² is true.

Hence, R is reflexive where (a, b) ∈ R

We know that a² = b² and b² = a² where (b, a) ∈ R

Hence, R is symmetric.

If (a, b) ∈ R and (b, c) ∈ R

We know that a² = b² and b² = c² we get a² = c² where (a, c) ∈ R

Hence, R is transitive.

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