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Question

# If X={1,2,3,4,5} and Y={1,3,5,7,9}, determine which of the following sets represent a relation and also a mapping?

A
R1={(x,y):y=x+2,xY,yY}
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B
R2={(1,1),(1,3),(3,5),(4,7),(5,9)}
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C
R3={(1,1),(2,3),(3,5),(3,7),(5,7)}
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D
R4={(1,3),(2,5),(4,7),(5,9),(3,1)}
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Solution

## The correct option is D R4={(1,3),(2,5),(4,7),(5,9),(3,1)}Here we will check all options one by one:Option A:R1={(x,y):y=x+2,x∈X,y∈Y}⟹R1={(1,3),(2,4),(3,5),(4,6),(5,7)}Since 4 amd 6 are the images of 2 and 4 respectively but 4 and 6 do not belong to Y∴(2,4),(4,6)∉X×YHence R1 is not a relation as well as not a mapping.Option B:R2: It is a relation but not a mapping because the element 1 has two different images.Option C:R2: It is a relation but not a mapping because the element 3 has two different images.Option D:R4: It is both a relation and a mapping because every element in X is mapped to the elements in Y. Also, every element of X has a one and only one image in Y and every element in Y has its pre-image in X. Hence, it is also one-one and onto mapping and hence it is a bijection.

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