# Practice for Third Midterm Exam

### Directions:

• Answer the problems on the exam pages.
• There are five problems for 100 total points. Scale will be determined after the exam but a good guess is A = 90, C = 60.
• 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) the power set of a set X

• (b) the codomain of a function

• (c) a one-to-one function

• (d) an antisymmetric binary relation

• (e) the rule of sums with overlap

• Question 2 (15): Determine whether each of these five statements is true or false. No explanation is needed or wanted, and there is no penalty for guessing.

• (a) If f(x) ≠ f(y), we know that f is not an onto function.

• (b) Let m and n be any two positive integers. Then there exist two sets A and B, where A has m elements, B has n elements, and A ∩ B has m + n - 2 elements.

• (c) There are more than 10 binary strings of length 4 that have at least two 1's.

• (d) If S is a set of seven dogs, and each dog is of exactly one breed from the set {retriever, spaniel, terrier}, then the must be at least two dogs of each breed.

• (e) For any finite set S, if P is the power set of S, then S ⊆ P.

• Question 3 (35): Let X be the set of dogs {Arly, Baxter, Cardie, Duncan, Ebony}. 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) {Y: Y ⊆ X and Y contains exactly two dogs}

• (b) {L: L is an ordered list from X of size 3}

• (c) {Z: L is an unordered list from X of size 4}

• (d) {Q: Q is a permutation from X of length 4, beginning with Baxter}

• (e) {(S, T): S and T are each sets of two dogs from X and S ∩ T = ∅}

• (f) {R: R is an permutation from X of size 3 where the dogs in R come in alphabetical order}

• (g) {P: P is a nonempty subset of X}

• Question 4 (15): Let A, B, and C be any three sets. Give an element-wise proof that (A - B) ∪ (B - C) ⊆ (B ∩ C)'. (Recall that if X and Y are any two sets, Y' is the complement of Y and "X - Y" means "(X ∩ Y')".)

• Question 5 (20): Let f: A → B and g: B → C be two functions that are each bijections. Let h: A → C be defined so that for any x in A, h(x) = g(f(x)). Prove that h is a bijection. (Recall that a function is a bijection if and only if it is both one-to-one and onto.)