Let R be a relation on the set A of ordered pairs of positive integers defined by (x,y)R(u,v) if and only if xv=yu. Show that R is an equivalence relation.
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Step-by-Step Solution
Step 1: Reflexivity
To prove reflexivity, we need to show that for any element (x, y) in the set A, (x,y)R(x,y) holds. According to the definition of the relation R, this means we need to check if xy=yx.
Step 2: Symmetry
To prove symmetry, we assume that (x,y)R(u,v) is true, which means xv=yu. We then need to show that (u,v)R(x,y) is also true. This requires demonstrating that uy=vx. Since multiplication of integers is commutative, xv=yu can be rewritten as vx=uy, which is equivalent to uy=vx. Thus, symmetry holds.
Step 3: Transitivity
To prove transitivity, we assume (x,y)R(u,v) and (u,v)R(a,b) are true. This means xv=yu and ub=va. Since x, y, u, v, a, b are positive integers, we can divide by them. From xv=yu, we get yx=vu. From ub=va, we get vu=ba. Combining these, we have yx=ba, which implies xb=ya. This shows that (x,y)R(a,b), thus transitivity holds.