Problem 3 (Properties of functions) (25 marks) Let A, B be two sets.
Let f be a function from A to B and let g be a function from B to A.
For each of the following functions F1, F2, F3, F4, F determine whether the
stated property is true or false. Justify your answer, hence, if the property
is true then prove it, otherwise, disprove it.
1. The property: if f and g are injective then Fi is injective, where Fi is
given by:
F1 :
A
x
g(f(x))
2. The property: if f and g are injective then F2 is injective, where F2 is
given by:
F2 :
AXB
B X A
(x, y)
(f(x), g(y))
3. The property: F3 is bijective, where F3 is given by:
F3 :
AXB
B XA
(x,y) > (y,x)
4. The property: if f and g are injective then F4 is bijective, where FA is
given by:
FA :
AXB - f(A) x 9(B)
(x,y)
(f (x), g(y))
5. The property: if f and g are bijective then F is bijective, where Is is
given by:
F5 :
A
f-1(9-1(x))