Question

Activity 10.5.

Suppose A is a set that definitely does not contain any cats, and let

f:P(A)→P(A∪{Grumpy Cat})

represent the function defined by

f(X)=X∪{Grumpy Cat}

(a) Verify that f is injective.

(b)Verify that f is not surjective.

(c) Describe specifically how to restrict the codomain of f to make it bijective.

restricting the codomain

the “induced” function X→B created from function f:X→Y and subset B⊆Y by “forgetting” about all elements of Y that do not lie in B, where B must contain the image of f

(d) Describe the input-output rule for f−1, the inverse function of the bijective version of f from Task c.

Answer 1

a)

But this means as A and B both don't contain Grumpy Cat

Thus, f must be injective

b) always contains the element "Grumpy Cat" so that is impossible as the left side always contains Grumpy Cat and the right side does not contain Grumpy Cat

c) We can restrict the co-domain to only contain sets that have "Grumpy Cat" therefore making the given function surjective

d) is the rule

Hope this was helpful. Please do leave a positive rating if you liked this answer. Thanks and have a good day!