I need help with this problem could someone solve and show m

I need help with this problem. could someone solve and show me how

thank you

18. Explain why Theorem (a) is equivalent to f^-1 is not a function if and only if f is not one to one 19. Let f: A -- > B and g: B -- > A Verify that g = f^-1 21. Let f be a function from A to B. Find f^-1

Solution

a)
f(x) = X\' = g(X)

P(S) is a power set which has all the elements of set

the compliment of every element of power set is also in the power set

the inverse of all the elemnts of f(X) is in g(X)

hence g= f^-1

b)

f^-1: B -> A

f^-1 = {(4,1) ,(1,2) ,(2,3) ,(3,4)}

so f^-1 =g