Exam 2MAT 200

Due Date: Nov 19, 2012, 5:30 pm.

Directions:

1. Everyone is expected to work on these problems without outside help (with exceptionof the textbook).

2. Students are encouraged to work together, but each student must write their own solu-tions independently. On your exam, write the names of the people you worked with.

3. Each problem is worth 10 points. Only do 6 out of the 7 problems. Clearly indicatewhich of the 6 problems you want graded by writing the problem numbers on the frontcover of your exam. If not clearly stated, only the first 6 problems will be graded.

4. For each of the following questions, give a full solution, not just an answer. Answerswithout justification will get little or no credit.

5. You must use the definitions in your textbook or the ones given below.

6. You must clearly cite any formulas/statements you use from the textbook.

7. You may use any formulas/statements you derive or prove within this exam.

8. You may NOT use any formulas/statements that you did not derive or are not fromthe textbook.

9. Late exams will NOT be graded.

Definitions:

• Let f : X → Y . g : Y → X is a left inverse of f if g ◦ f = IX , where IX stands for theidentity function on X.

• Sur(X, Y ) = {f ∈ Fun(X, Y ) | f is surjective}.

Problems:

1. Let f : X → Y , where X and Y are non-empty sets.

(a) (3 points) Prove that f is injective if and only if f has a left inverse.

(b) (3 points) Suppose that g : Y → X is a left inverse to f . Prove that g is surjective.

(c) (4 points) Can two different injections share a common left inverse? If yes, give anexample. If not, give a proof.

2. Let f : X → Y be injective, and |X| = m, |Y | = n for some positive integers m and n.

(a) (5 points) Find the number of left inverses f has. Give an informal argument.

(b) (5 points) Using induction, prove that the formula you found in part (a) is correct.

3. Let |X| = m for some positive integer m and Y = {a, b, c}. For each y ∈ Y , defineAy = {f ∈ Fun(X, Y ) | y is not a value of f}.(a) (3 points) Find |Aa|, |Aa ∩ Ab|, and |Aa ∩ Ab ∩ Ac|.

(b) (4 points) Prove that Sur(X, Y ) = Fun(X, Y )− (Aa ∪ Ab ∪ Ac).

(c) (3 points) Compute |Sur(X, Y )|. (Hint: Use some form of the exclusion-inclusionprinciple.)

4. Suppose that you are given 20 distinct positive integers less than 96.

(a) (5 points) Show that there are two disjoint subsets (of the 20 integers) with thesame sum.

(b) (5 points) Show that there are two disjoint subsets (of the 20 integers) of cardinality2 with the same sum. (Hint for both parts: the pigeonhole principle.)

5. Refer to the textbook for the exact definitions.

(a) (2 points) What does |X| ≤ |Y | mean for finite sets? For |X| = |Y |? (Hint: Are|X| and |Y | numbers?)

(b) (4 points) What does |X| ≤ |Y | mean for infinite sets? For |X| = |Y |? Explainhow the definitions for the infinite sets are consistent with the definitions for finitesets.

(c) (4 points) Is it possible for a powerset P(X) to be denumerable? If yes, give anexample. If not, give a proof.

6. Let X, Y and Z be infinite sets. Prove the following:

(a) (2 points) |X| = |X|.(b) (2 points) If |X| = |Y |, then |Y | = |X|.(c) (3 points) If |X| = |Y | and |Y | = |Z|, then |X| = |Z|.(d) (3 points) If |X| ≤ |Y | and |Y | ≤ |Z|, then |X| ≤ |Z|.

7. Let M be the set of real roots of polynomials of the form x2 + bx + c, where b and c areintegers. (Note: b and c are not fixed integers.)

(a) (3 points) Prove that Z ⊆M , but Z 6= M .

(b) (4 points) Is Q ⊆M? Is M ⊆ Q? Prove or give a counterexample.

(c) (3 points) Prove that M is a denumerable set.