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Question
Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.
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Solution
We observe the following properties of R.
Reflexivity : Let (a, b) be an arbitrary element of the set A. Then,
(a, b) ∈ A
⇒ ab = ba
⇒ (a, b) R (a, b)
Thus, R is reflexive on A.
Symmetry : Let (x, y) and (u, v)∈A such that (x, y) R (u, v). Then,
xv=yu
⇒ vx=uy
⇒ uy=vx
⇒ (u, v) R (x, y)
So, R is symmetric on A.
Transitivity : Let (x, y), (u, v) and (p, q)∈R such that (x, y) R (u, v) and (u, v) R (p, q)
⇒ xv = yu and uq = vp
Multiplying the corresponding sides, we get
xv × uq = yu × vp
⇒ xq = yp
⇒ (x, y) R (p, q)
So, R is transitive on A.
Hence, R is an equivalence relation on A.
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