# Third Midterm Exam

### Directions:

• Answer the problems on the exam pages.
• There are five problems for 100 total points. Actual scale was A = 85, C = 55.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

```  Q1: 15 points
Q2: 15 points
Q3: 35 points
Q4: 15 points
Q5: 20 points
Total: 100 points
```

• Question 1 (15): Briefly identify the following terms or concepts (3 points each):

• (a) an equivalence relation

• (b) for two sets to be disjoint

• (c) what it means for a relation R ⊆ A × B to be a function

• (d) a bijection

• (e) the Product Rule for counting sets

• Question 2 (15): Determine whether each of these five statements is true or false (3 points each). No explanation is needed or wanted, and there is no penalty for guessing. In the first three statements, A and B are finite sets and f: A → B is a function.

• (a) If B has more elements than A, then f is not onto.

• (b) If f is not onto, then B has more elements than A.

• (c) If A has m elements and B has n elements, then the power set of A × B has 2m × 2n elements.

• (d) If I distribute seven treats among four dogs, then at least one dog gets more than two treats.

• (e) Of all the binary strings of length 5, more have an odd number of ones than have an even number of ones.

• Question 3 (35): Let X be the set of dogs {Arly, Baxter, Cardie, Duncan}. Give the sizes of each of the following sets. No proof is necessary for a correct answer, but a justification may help with partial credit (5 points each).

• (a) {F: F is a subset of X containing Baxter but not Duncan}

• (b) {G: G is an ordered list from X of size 5}

• (c) {H: H is an unordered list from X of size 7}

• (d) {I: I is a permutation from X of length 3, not containing Baxter}

• (e) {J: J is a subset of X and the size of J is odd}

• (f) {K: K is a binary relation on X (a subset of X × X)}

• (g) {L: L is a permutation form X of size 5}

• Question 4 (15): Let A, B, C, and D be any four sets. Assume that A ⊆ B and that C ⊆ D. Give an element-wise proof that (A ∪ C) ⊆ (B ∪ D).

• Question 5 (20): Let A, B, and C be finite sets with |A| > |B| and |A| = |C|. (Here "|X|" denotes the number of elements in X.) Let f: A → B and g: B → C be any two functions. Define h: A → C be defined by the rule h(a) = g(f(a)). Prove that h not a bijection.

Last modified 19 November 2015