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Which of the following is a mapping or total function

1) f is defined on RXR

f(x)=1    ;x=5t, t ℇ Z

    =0    ;x=5t+1, t ℇ Z

2)1) f is defined on RXZ

f(x)=1    ;x=5t, t ℇ Z

    =0    ;x=5t+1, t ℇ Z

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We say a function f : A --> B i.e. function from A to B is a total function or mapping if the domain of the function f is equal to the cardinality of set A.

In the first example we are given f(x) so x should be considered for domain. Given x = 5t where t is given that it belongs to integers (Z).So x also belongs to integers only and not real numbers

But f is defined from R × R, so cardinality of A set in given A --> B is the set of real numbers.

Hence the domain does not cover all elements of A set. Since only specific integers are covered only. Hence the given function is a partial function not a total function.

Similarly the 2nd set is also defined on R × Z but the domain is the set of integers only. Hence it is not total function as well.

Had the function been defined Z × R or Z ×  Z then the function would be total function.
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