Hello, I am taking a Discrete Math class and am having a really hardtime. If you

Question

Hello,

I am taking a Discrete Math class and am having a really hardtime. If you can please try to explain in laimants terms thefollowing formula. Much appriciated.

Requirement: "show that the given relation R is an equivalencerelation on set S. Then describe the equivalence class containingthe given element z in S, and determine the number of distinctequivalence classes of R".

Pproblem: Let S be the set of ordered pairs of positiveintegers, let z = (5,8), and define R so that (x1,x2) R (y1,y2)means that x1 + y2 = y1 + y2

The x and y with numbers mean that is power 1 and 2.

Thank you very much. If you can also recommend an easy wayto get through this class I would appriciate.

Explanation / Answer

  and the relation is defined as (a,b) R (c,d)if a+d = c+d.
R is reflexive:    (a,b) R (a,b), since a+b = a+b R is symmetric:    sinceif  (a,b) R (c,d) implies a+d = c+d then a=c. Hence c+ b = a +b.       That is (c,d) R (a,b) R is transitive:    Suppose (a,b)R(c,d) and (c,d)R (e,f)    That isa+d = c+d and c+f = e+f.    So a=cand c = e and hence a+f = c+f = e+f.    Thisimplies (a,b)R(e,f).   So we proved that R is an equivalencerelation.
  The equivalence class containig (5,8) = [(5,8)] ={ (a,b) in S | (5,8) R (a,b) }               = { a,b) in S | (5+b) = a+b } = {a,b) in S | a=5 } = { (5,b) | b positiveintegers}

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