Let A {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}, and define R on A by (x1, y1) R (x2, y2)

Question

Let A {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}, and define R on A by (x1, y1) R (x2, y2) if x1 + y1 x2 + y2.

a) Verify that 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

for A and R to be an equivalence relation its reflexive: (x,y)R(x,y) as x+y=x+y its symmetric: (x,y)R(c,d) => (c,d)R(x,y) as x+y=c+d means c+d=x+y transitive : (x,y)R(c,d) and (c,d)R(e,f) => (x,y)R(e,f) as x+y=c+d=e+f Equivalence classes for: [(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: partition: [(1,1)] [(1,2),(2,1)] [(1,3),(2,2),(3,1)] [(1,4),(2,3),(3,2),(4,1)] [(1,5),(2,4),(3,3),(4,2),(5,1)] [(2,5),(3,4),(4,3),(5,2)] [(3,5),(4,4),(5,3)] [(4,5),(5,4)] [(5,5)] message me if you have any doubts

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