show that each of the relation R in the set A={x belongs to z:0 is less than or equal to x is less than or equal to 12} given by a. R={(a, b):|a-b|is a multiple of 4} b. R={(a, b):a=b}is an equivalence relation.
A= { x belongs to Z:0 <= x <=12}
A= { 0,1,2,3,4,5,6,7,8,9,10,11,12}
R= {(a,b) :|a-b| is a multiple of 4}
For a belongs to A and (a,a) belongs to R as |a-a|=0 which is a multiple of 4.
Hence it is reflexive.
Let (a,b) belongs to R and |a-b| is a multiple of 4
then |b-a| is also a multiple of 4.
Hence it is symmetric.
Let (a,b) belongs to R and |a-b| is a multiple of 4 and (b,c) belongs to R and |b-c| is also a multiple of 4,
then (a-c) = (a-b) + (b-c) is also a multiple of 4
This implies that (a-c) is also a multiple of 4
and also (a,c) belongs to R
Hence it is transitive.
This three criteria implies that the relation is a equivalence relation.
Hope this helps you.
All the best !!
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