Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_relations

Class 12 Maths · Relations And Functions · Very Long Answer #226

On N={1,2,…}, let R={(x,y):2x+y=41}. Find its domain and range and test the three relation properties.

✅ Step-by-Step Solution

Final Answer: Domain {1,…,20}; range {1,3,…,39}; none of reflexivity, symmetry, transitivity.

Positive integer coordinates require y=41−2x≥1, giving x=1,…,20.
The corresponding y values are 39,37,…,1. Thus the domain is {1,…,20} and the range is the odd integers from 1 to 39.
A diagonal pair would require 3x=41, which has no natural solution; hence reflexivity fails. Also (1,39) occurs but (39,1) does not, so symmetry fails.
The pairs (11,19) and (19,3) satisfy the rule, but (11,3) does not because 2·11+3=25≠41. Thus transitivity fails.
The domain and range are as stated, and R is neither reflexive, symmetric nor transitive.

[1 mark] Positive integer coordinates require y=41−2x≥1, giving x=1,…,20.
[1 mark] The corresponding y values are 39,37,…,1. Thus the domain is {1,…,20} and the range is the odd integers from 1 to 39.
[1 mark] A diagonal pair would require 3x=41, which has no natural solution; hence reflexivity fails. Also (1,39) occurs but (39,1) does not, so symmetry fails.
[1 mark] The pairs (11,19) and (19,3) satisfy the rule, but (11,3) does not because 2·11+3=25≠41. Thus transitivity fails.
[1 mark] The domain and range are as stated, and R is neither reflexive, symmetric nor transitive.

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