**Problem 1:**

Prove that a function f is 1-1 iff for all . Given that .

**Problem 2:**

Prove that a function if is onto iff for all . Given that .

**Problem 3:**

(a) How many functions are there from a non-empty set S into \?

(b) How many functions are there from into an arbitrary set ?

(c) Show that the notation implicitly involves the notion of a function.

**Problem 4:**

Let be a function, let , , and . Prove that

i)

ii)

iii)

**Problem 5:**

Let I be a non-empty set and for each , let be a set. Prove that

(a) for any set B, we have

(b) if each is a subset of a given set S, then where the prime indicates complement.

**Problem 6:**

Let A, B, C be subsets of a set S. Prove the following statements:

(i)

(ii)

đ đ đ

Nalin Pithwa

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