1. Let f : A B be a function. Let S and T be subsets of A. Prove that f(S T) f(S) f(T). Give an example where f(S T) 6= f(S) f(T).

2.Let f : A B and g : B C be two functions. Let h = g f : A C. Prove thatIm(h) Im(g). Give an example where Im(h) 6= Im(g). Here Im(h) denotes the image of function h.

3.A four-digit number is a number which is represented in decimal system by four digits with the first digit not equal to zero. How many four-digit numbers can be made out of digits 0, 1, 2, 3, 8, 9 if

1. the digits in a number can repeat;

2. no digit can repeat;

3. the number is odd and digits can repeat;

4. the number is odd and digits cannot repeat. Explain your answers.

4. Let A and B be two sets such that |A| = |B| = n. Find the number of bijections from A to B. Explain.

5. A word is a string of letters from an alphabet. How many words of length l can be formed from an alphabet of size m? How many can be formed in which each letter is used at most once? Explain.