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Discrete mathematics exercises
The equivalence relation only needs to be proved to satisfy reflexivity, symmetry and transitivity.

The reflexivity is obvious (in the relationship, X and Y are replaced by U and V respectively, which can be proved)

Symmetry:

because

x+v=u+y

& ltx,y & gtR & ltu,v & gt

Transitivity:

& ltu,v & gtR & ltx,y & gt? u+y=x+v

& ltx,y & gtR & lta,b & gt? x+b=a+y

rule

(u+y)+(x+b)=(x+v)+(a+y)

u+b=a+v

& ltu,v & gtR & lta,b & gt