Question

Let Z be the set of integers. Show that the relation R={(a,b):a,b∈Z and a+b is even } is an equivalence relation on Z.

Answer 1

Answer:

Step-by-step explanation:

The given relation R on the set of integers Z is reflexive as (x,x)∈R as 2x is even for all x∈Z.

Again the relation is symmetric as if (x,y)∈R⇒(y,x)∈R since x+y is even gives y+x is also even for all x,y∈Z.

Again this relation R is transitive as if (x,y)∈R and (y,z)∈R this gives (x,z)∈R as x+y=2k  

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This gives x+z is even.

Hence this relation is also transitive.

So the relation is equivalance relation.