Let Z be the set of integers. Show that the relation R={(a,b):a,b∈Zand a+b is even } is an equivalence relation on Z.
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Solution
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=2k1.....(1) [ Since x+y is even ] and y+z=2k2.....(2) [ Since y+z is even] [ For k1 and k2 being integers]