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
The Answer to the Question
is below this banner.
Here's the Solution to this Question
A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and
I. Reflexive: Let be an ordered pair of integers, To show C is reflexive we must show Multiplication of integers is commutative, so Thus
II. Symmetric: Let and be ordered pairs of integers such that Then This equation is equivalent to so This shows is symmetric.
III. Transitive: Let and be ordered pairs of integers such that and . Then and Thus, and which implies Since we can cancel it from both sides of this equation to get This shows and so is transitive.
Since is reflexive, symmetric, and transitive then is an equivalence relation.