Question

let A be the set of integers and C be the set of ordered pairs (x,y)£A×A such that y is not equal to zero define relation ~ on C(x,y)~(z,w) if yz=zw prove that defines an equivalence relation on C

Answer 1

A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and

transitive.

I. Reflexive: Let $(x, y)$ be an ordered pair of integers, $y\not =0.$To show C is reflexive we must show $((x, y),(x,y))\in C.$ Multiplication of integers is commutative, so $xy=yx.$ Thus $((x, y),(x,y))\in C.$

II. Symmetric: Let $(x, y)$and $(z,w)$ be ordered pairs of integers such that $((x, y),(z,w))\in C.$ Then $yz=xw.$ This equation is equivalent to $wx=zy,$ so $((z, w),(x,y))\in C.$ This shows $C$ is symmetric.

III. Transitive: Let $(x,y), (z,w),$ and $(u,v)$ be ordered pairs of integers such that $((x, y),(z,w))\in C$ and $((z, w),(u,v))\in C$. Then $yz=xw$ and $wu=zv.$ Thus, $yzu=xwu$ and $xwu=xzv,$ which implies $yzu=xzv.$ Since $z\not=0,$ we can cancel it from both sides of this equation to get $yu=xv.$ This shows $((x, y),(u,v))\in C,$ and so $C$ is transitive.

Since $C$ is reflexive, symmetric, and transitive then $C$ is an equivalence relation.