Class 12 Relations And Functions · Long Answer3 Markstypes_of_relations

Class 12 Maths · Relations And Functions · Long Answer #176

On A×A, A={1,…,9}, let (a,b)R(c,d) iff a+d=b+c. Prove equivalence and find [(2,5)].

✅ Step-by-Step Solution

Final Answer: Equivalence; [(2,5)]={(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}.

Rewrite the predicate as a−b=c−d. Each pair has the same difference as itself, proving reflexivity.
Equality is symmetric; equality of the differences for first/second and second/third pairs implies equality for first/third, proving symmetry and transitivity.
For (x,y) in [(2,5)], x−y=2−5=−3, so y=x+3. Requiring x,y∈A gives x=1,…,6.
R is an equivalence relation and the class is {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}.

[0.5 marks] Rewrite the predicate as a−b=c−d. Each pair has the same difference as itself, proving reflexivity.
[1 mark] Equality is symmetric; equality of the differences for first/second and second/third pairs implies equality for first/third, proving symmetry and transitivity.
[1 mark] For (x,y) in [(2,5)], x−y=2−5=−3, so y=x+3. Requiring x,y∈A gives x=1,…,6.
[0.5 marks] R is an equivalence relation and the class is {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}.

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