Solution to Consider the following relations on the set A={1,2,3,4,5}. a) R1={(1,1), (1,2), (3,3), (3,2)} b) R2={(1,1), … - Sikademy
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Archangel Macsika

Consider the following relations on the set A={1,2,3,4,5}. a) R1={(1,1), (1,2), (3,3), (3,2)} b) R2={(1,1), (2,2), (3,2), (4,5), (5,2)} c) R3={(1,4), (2,5), (3,3), (4,2), (5,2)} State, with reasons, for a), b), c) whether the relation is: I) a function from A to A ii) an onto function iii) a one-to-one function iv) an everywhere defined function

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(i). For (a),

Rwill not be a function from A to A as there is more than one image of 1 and 3 in R1 (i.e (1, 1), (1, 2), (3, 3), (3, 2)) which is not allowed in a function.

For (b),

R2 will be a function as there is no unmatched element in the domain of R2 and there is only one image of all the elements of domain.

For (c),

R3 will be a function as there is no unmatched element in the domain of R3 and there is only one image of all the elements of domain.


(ii). For (a),

Rwill not be a function.

For (b),

R2 will not be an onto function as 3 and 4 elements in the range are not matched with any element of the domain, which is not a property of an onto function.

For (c),

R3 will not be an onto function as 1 element in the range is not matched with any element of the domain, which is not a property of an onto function.


(iii). For (a),

Rwill not be a function.

For (b),

R2 will not be a one-to-one function as 2, 3 and 5 have multiple images in the codomain which is not allowed in a one-to-one function.

For (c),

R3 will not be a one-to-one function as 4 and 5 have multiple images in the codomain which is not allowed in a one-to-one function.


(iv). For (a),

Rwill not be a function.

For (b),

R2 will be an everywhere defined function, as the domain of function is the domain itself.

For (c),

R3 will be an everywhere defined function, as the domain of function is the domain itself.


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