Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_relations

Class 12 Maths · Relations And Functions · Very Long Answer #222

On A×A, A={1,…,10}, define (a,b)R(c,d) iff a+d=b+c. Prove equivalence and find [(3,4)].

✅ Step-by-Step Solution

Final Answer: Equivalence; [(3,4)]={(1,2),(2,3),…,(9,10)}.

Rewrite the predicate as a−b=c−d. Equality of each difference with itself gives reflexivity.
If a−b=c−d, then c−d=a−b; hence R is symmetric.
If a−b=c−d and c−d=e−f, then a−b=e−f, proving transitivity and thus equivalence.
For (x,y)∈[(3,4)] the condition is x−y=3−4=−1. Thus y=x+1, with x=1,…,9 to keep both coordinates in A.
The equivalence class is {(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9),(9,10)}.

[1 mark] Rewrite the predicate as a−b=c−d. Equality of each difference with itself gives reflexivity.
[1 mark] If a−b=c−d, then c−d=a−b; hence R is symmetric.
[1 mark] If a−b=c−d and c−d=e−f, then a−b=e−f, proving transitivity and thus equivalence.
[1 mark] For (x,y)∈[(3,4)] the condition is x−y=3−4=−1. Thus y=x+1, with x=1,…,9 to keep both coordinates in A.
[1 mark] The equivalence class is {(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9),(9,10)}.

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