Show that the relation R = {(a, b) ∈ ℤ × ℤ: |a| = |b|} is an equivalence relation on ℤ.

Answer 1

Reflexive:

| (a,b) | = sqrt( a^2+b^2) = | (a,b) |

Symmetric:

| (a,b) | = sqrt( a^2 + b^2) = sqrt(b^2 + a^2) = | (b,a) |

Transitive:

Given: (a,b) ~(c,d) and (c,d) ~ (e,f)

Then sqrt(a^2+b^2) = sqrt(c^2+d^2) and sqrt(c^2+d^2) = sqrt(e^2+f^2)

sqrt(a^2+b^2) = sqrt(e^2+f^2) <-- transitive / substitution property

|(a,b)| = |(e,f)|

(a.b)~(e,f)