# CMPSCI 190DM: A Mathematical Foundation for Informatics

# Practice for Third Midterm Exam

#### David Mix Barrington

#### 12 November 2015

### 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.)
Last modified 12 November 2015