Let R be a relation on the set A of ordered pairs of non-zero integers defined by (x,y)R(u,v). If xv=yu, then show that R is an equivalence relation.

Question

Filo · Accepted Answer

The relation R on Z×Z is defined by(x,y)R(u,v)⇔xv=yu for all (x,y),(u,v)∈Z×ZWe observe the following properties of R on Z×ZReflexivity: For any (x,y)∈Z×Zxy=yx⇒ (x,y)R(x,y)Thus, (x,y)R(x,y) for all (x,y)∈Z×ZSo, R is a reflexive relation on Z×ZSymmetry: Let (x,y),(u,v)∈Z×Z such that (x,y)R(u,v). Then, (x,y)R(u,v)⇒xv=yu⇒uy=vx⇒(u,v)R(x,y)Thus, (x,y)R(u,v)⇒(u,v)R(x,y) for all (x,y),(u,v)∈Z×ZSo, R is a symmetric relation on ZTransitivity: Let (x,y),(u,v),(a,b)∈Z×Z be such that (x,y)R(u,v) and (u,v)R(a,b)Then,(x,y)R(u,v)⇒xv=yuand (u,v)R(a,b)⇒ub=va⇒ (xv)(ub)=(yu)(va)⇒xb=ya⇒(x,y)R(a,b)So, R is a transitive relation on Z×ZHence, R is an equivalence relation on Z×Z

Question Text	Let $R$ be a relation on the set $A$ of ordered pairs of non-zero integers defined by $(x, y) R (u, v)$ . If $xv = y u$ , then show that $R$ is an equivalence relation.
Answer Type	Text solution:1
Upvotes	150

Let $R$ be a relation on the set $A$ of ordered pairs of non-zero integers defined by $(x, y) R (u, v)$ . If $xv = y u$ , then show that $R$ is an equivalence relation.

Text solutionVerified

Practice questions from similar books