Let A = {1, 2, 3, 4, 5} X {1, 2, 3, 4, 5}, and define R on A by (x1, y1) R (x2,

Question

Let A = {1, 2, 3, 4, 5} X {1, 2, 3, 4, 5}, and define R on A by (x1, y1) R (x2, y2) if x1 + y1 = x2 + y2. a) Verify that R is an equivalence relation on A. b) Determine the equivalence classes [(1, 3)], [(2, 4)], and [(1, 1)]. c) Determine the partition of A induced by R.

Explanation / Answer

a) let (x,y) belong to A.. then x+y=x+y always.. hence (x,y) R (x,y).. hence R is reflexive.. if (x1, y1) R (x2, y2) => x1+y1= x2+y2 => x2+y2= x1+y1 =>(x2,y2) R (x1,y1) hence R is symmetric if (x1, y1) R (x2, y2) and (x2, y2) R (x3, y3) => x1+y1= x2+y2= x3+y3 hence (x1, y1) R (x3, y3) hence R is transitive.. so r is an equivalence relation.. b) [(1,3)]= {(1,3),(2,2),(3,1)} [(2,4)]={(1,5),(2,4),(3,3),(4,2),(5,1)} [(1,1)]={(1,1)} c) A is partitioned into the equivalence classes by R hence the partition of A is A is the union of {[(1,1)],[(1,2)],[(1,3)],[(1,4)],[(1,5)],[(2,5)],[(3,5)],[(4,5)],[(5,5)]} PLEASE RATE MY ANSWER FIRST>.:)

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