Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_relations

Class 12 Maths · Relations And Functions · Very Long Answer #180

On N×N, define (a,b)R(c,d) iff ad(b+c)=bc(a+d). Prove that R is an equivalence relation.

✅ Step-by-Step Solution

Final Answer: An equivalence relation.

Divide by the positive product abcd to obtain 1/b+1/c=1/d+1/a, or 1/a−1/b=1/c−1/d. Put h(a,b)=1/a−1/b.
For any pair p, h(p)=h(p), so pRp: reflexivity holds.
If pRq, then h(p)=h(q); reversing this equality proves qRp, so symmetry holds.
If pRq and qRr, then h(p)=h(q)=h(r), giving pRr and proving transitivity.
The three properties hold, so the original relation is an equivalence relation.

[1 mark] Divide by the positive product abcd to obtain 1/b+1/c=1/d+1/a, or 1/a−1/b=1/c−1/d. Put h(a,b)=1/a−1/b.
[1 mark] For any pair p, h(p)=h(p), so pRp: reflexivity holds.
[1 mark] If pRq, then h(p)=h(q); reversing this equality proves qRp, so symmetry holds.
[1 mark] If pRq and qRr, then h(p)=h(q)=h(r), giving pRr and proving transitivity.
[1 mark] The three properties hold, so the original relation is an equivalence relation.

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