Define F ga b c rightarrow Z as follows For all A in ga b c

Define F: g({a, b, c}) rightarrow Z as follows: For all A in g({a, b, c}): F(A) = the number of elements in A. Is F one-to-one? Prove or give a counterexample. Is F onto? Prove or give a counterexample.

Solution

Here a function is 1-1 if each element of domain is associated with one and only one element of its range.

Now here given domain is a power set of three elements a,b and c, so A may be any its subset as {a}, {b},{c},{ab},{ac},{bc},{a,b,c} and a null set.

Then by definition, F( {a,b}) = 2 as it has two elements.

Also F({ac} =2

that means for two different values of domain, they have the same value in range that means F is not 1-1 ever.

This is the answer of part (a).

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Now we have that as the range of F is given as Z that means in its codomain , there will be some negative integers also

that can not have any preimage in its domain as domain is the set of countable numbers. Thus clearly as few elements of codomain are not image of its domain, that means F is not ONTO.

This is a counterexample also.

This is the answer of part (b)