Midterm Practice Problems MTH 231
The following are practice problems from each of the four Modules of material we have covered so far. They are organized by learning objective. It is suggested that students create their own “practice midterms” by selecting 1-3 problems from each Module and then try to complete the practice midterm within the allotted time (90 minutes to take the midterm and 20 minutes to scan and upload your midterm solutions to Gradescope). Do not limit your study to just these exercises; it is recommended that students also review the Instructor Notes and retry previously missed homework problems. There are also additional exercises available in the Epp and Rosen texts (which students may access through the OSU library website).
1 PROPOSITIONALLOGIC
1.1
1.
2.
1.2
3.
4. 5.
1.3
6.
1.4
7.
IDENTIFY SENTENCES THAT ARE PROPOSITIONS AND NON-PROPOSITIONS.
Determine if each sentence is a true proposition, a false proposition, or not a proposition.
a. 𝑥2≥0.
b. Are you in CS 161?
c. (5𝑥−1=19) → (𝑥=4).
d. If an animal is a horse, then it is a reptile.
e. If the flying spaghetti monster exists, then 1+2=3.
f. Wear your seat belt.
Give an example of two propositions 𝑝 and 𝑞 such that 𝑝 ∨ 𝑞 is true but 𝑝 ∧ 𝑞 is false. What can you then conclude about the truth value of 𝑝 ↔ 𝑞?
DETERMINE TRUTH TABLES OF COMPOUND PROPOSITIONS.
Let 𝑝, 𝑞, and 𝑟 be propositions. Write a truth table for the following compound propositions.
a. 𝑝⊕(𝑝→𝑞)
b. ¬(𝑝→𝑞)∧(¬𝑞∨𝑟)
Use a truth table or logical equivalences to determine if (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ ¬𝑞) → 𝑝 is a tautology, contradiction, or contingency.
Show that (𝑝 → 𝑟) ∧ (𝑞 → 𝑟) and (𝑝 ∨ 𝑞) → 𝑟 are logically equivalent. NEGATE COMPOUND PROPOSITIONS USING DEMORGAN’S LAWS.
Negate the following compound propositions, where 𝑝, 𝑞, and 𝑟 are propositions.
a. MTH 112 or MTH 151 are required prerequisites to take this course.
b. To ride the roller coaster, you must be at least 48 inches tall or at least 16 years old.
c. ¬𝑝∨(𝑝∧𝑞)
d. (𝑝∧𝑞)∨(¬𝑞∧𝑟)
FORM CONTRAPOSITIVES OF IMPLICATIONS.
Write each of the following implications in the form “if 𝑝, then 𝑞.” Then write its contrapositive.
a. It is necessary to walk eight miles to get to the top of Long’s Peak.
b. Your guarantee is good only if you bought your smartphone 1 year ago.
Source: Rosen (2019). Discrete Mathematics and Its Applications, 8th Ed. McGraw-Hill.
Midterm Practice Problems
MTH 231
1.5
c. It snows whenever the wind blows from the northeast. d. A sum 𝑎 + 𝑏 is irrational if 𝑎 or 𝑏 are irrational.
DETERMINE THE TRUTH VALUE OF QUANTIFIED PROPOSITIONS. Determine the truth value of each of the following propositions.
a. ∃𝑛∈Zsuchthat𝑛>1 → 𝑛2 =𝑛.
b. ∃𝑛∈Zsuchthat(𝑛>1)∧(𝑛2 =𝑛).
c. ∀𝑥∈R,∃𝑚∈Zsuchthat𝑚>𝑥.
d. ∃𝑚∈Zsuchthat∀𝑥∈R,𝑚>𝑥.
e. ∃𝑚∈Zsuchthat∀𝑥∈R,𝑚≤𝑥2.
f. ∃𝑥,𝑦∈R(𝑥+𝑦=10)
g. ∀𝑥,𝑦∈R(𝑥+𝑦=10)
Give an example of a propositional function 𝑃(𝑥) such that ∃𝑥 𝑃(𝑥) is true but ∀𝑥 𝑃(𝑥) is false. (Make sure you specify the domain of 𝑃(𝑥).)
1.6
NEGATE QUANTIFIED PROPOSITIONS.
10. State the negation of each of the following propositions.
a. Every species of rhinoceros has gone extinct.
b. No one has read Spillover by David Quammen.
c. There is a student in this class who plays the flute.
11. Negate each of the propositions in problem 8 above.
8.
9.
2 SETTHEORY
2.1
APPLY UNION, INTERSECTION, SET DIFFERENCE, AND COMPLEMENT OPERATIONS.
12. Given the following sets 𝐴 and 𝐵, find 𝐴 ∪ 𝐵, 𝐴 ∩ 𝐵, 𝐴 − 𝐵, and 𝐴. Assume that the universal set 𝑈 is the set of prime numbers less than 20.
a. 𝐴 = {2,3,5,7}, 𝐵 = {11,13,17,19}
b. 𝐴 = ∅, 𝐵 = {11,13,17,19}
13. Assume that the universal set 𝑈 is the set of prime numbers less than 20. Find two sets 𝐴 and 𝐵 such that 𝐴 ∪ 𝐵 = 𝑈 and 𝐴 − 𝐵 = ∅. (More than one correct answer is possible.)
DETERMINE THE DIRECT PRODUCT OF TWO OR MORE SETS.
14. If 𝐴 = {𝑎, 𝑏, 𝑐}, 𝐵 = {𝑎, 𝑏}, and 𝐶 = {𝑏, 𝑐}, determine 𝐴 × 𝐵 and 𝐴 × 𝐵 × 𝐶.
15. Let 𝐴 = {1,2}, 𝐵 = {2,3}, and 𝐶 = {2,3,4}.
a. Find(𝐴∩𝐵)×𝐶and𝐴∩(𝐵×𝐶).
b. Is it true or false that whenever 𝐴, 𝐵, and 𝐶 are sets, (𝐴 ∩ 𝐵) × 𝐶 = 𝐴 ∩ (𝐵 × 𝐶)?
2.2
Source: Rosen (2019). Discrete Mathematics and Its Applications, 8th Ed. McGraw-Hill.
Midterm Practice Problems MTH 231
2.3
IDENTIFY FUNCTIONS AS SETS AND DETERMINE WHEN A FUNCTION IS ONE-TO-ONE AND/OR
ONTO.
16. Let 𝑓: Z → Z such that 𝑓(1) = 3, 𝑓(2) = 5, and 𝑓(3) = 7. Write the function 𝑓 as a set of
ordered pairs.
17. The following functions are expressed as ordered pairs. Determine if each function is one-to-one and/or onto. Assume that the codomain of each function is {1, 2, 3}.
a. 𝑓 = {(1,1), (2,2), (3,2)}
b. 𝑔 = {(1,2), (2,1), (3,3)}
18. If 𝑓: R → R, determine if each function is one-to-one and/or onto. What if 𝑓: Z → Z?
c. 𝑓(𝑥)=2𝑥+5
d. 𝑓(𝑥) = |𝑥|
e. 𝑓(𝑥) = ⌈𝑥⌉
DETERMINE THE POWER SET OF A SET.
19. Determine 𝒫(𝑆) for each of the following sets 𝑆.
a. 𝑆={𝑥}
b. 𝑆={𝑥,𝑦,𝑧}
20. Let 𝐴 and 𝐵 be sets. Determine if each of the following statements is true or false. If it is true, provide a proof. If it is false, provide a specific counterexample.
a. 𝒫(𝐴)∪𝒫(𝐵) ⊆ 𝒫(𝐴∪𝐵). b. 𝒫(𝐴∪𝐵) ⊆ 𝒫(𝐴)∪𝒫(𝐵).
DERIVE SOME BASIC COUNTING RULES FROM SET OPERATIONS.
21. At your upcoming family gathering you plan to serve two types of desserts: one type containing
dairy and gluten, and a nondairy gluten-free type. If you have 5 family members who avoid dairy, 6 family members who avoid gluten, and 3 family members who avoid both, how many nondairy gluten-free desserts will you need to prepare (assuming everyone else gets the dessert containing dairy and gluten)?
22. If the cardinality of a set 𝐴, |𝐴| = 𝑚, determine |𝐴𝑛|. (Here, 𝐴𝑛 = 𝐴 × 𝐴 × ⋯ × 𝐴 refers to the set of 𝑛-tuples where each term of the 𝑛-tuple is an element of 𝐴.
TAKE THE UNION AND INTERSECTION OF FAMILIES OF SETS.
23. Define the collection of sets 𝐴 = [−𝑛, 𝑛]1, where 𝑛 ∈ Z+. Find ∪∞ 𝐴 and ∩∞ 𝐴 .
2.4
2.5
2.6 2.7
𝑛 𝑛=1 𝑛 𝑛=1 𝑛
IDENTIFY FAMILIES OF SETS THAT ARE PAIRWISE DISJOINT.
24. Give an example of sets 𝐴, 𝐵, and 𝐶 such that 𝐴 ∩ 𝐵 ∩ 𝐶 = ∅ but 𝐴 ∩ 𝐵 ≠ ∅, 𝐵 ∩ 𝐶 ≠ ∅, and
𝐴 ∩ 𝐶 ≠ ∅.
1 This is the closed interval from −𝑛 to 𝑛.
Source: Rosen (2019). Discrete Mathematics and Its Applications, 8th Ed. McGraw-Hill.
Midterm Practice Problems MTH 231
25. Let 𝑈 = {1,2,3,4,5,6,7,8,9,10}. Give an example of pairwise disjoint sets 𝐴, 𝐵, 𝐶, and 𝐷 such that𝐴∪𝐵∪𝐶∪𝐷 = 𝑈.
2.8 USE VENN DIAGRAMS TO DETERMINE WHEN SETS ARE EQUAL OR NOT.
26. Let 𝐴 and 𝐵 be sets. Show that if 𝐴 ∪ 𝐵 = 𝐴 ∩ 𝐵, then 𝐴 = 𝐵. Illustrate using a Venn diagram.
27. Let 𝐴 and 𝐵 be sets. Show that 𝐴 ∩ 𝐵 = 𝐴 ∪ 𝐵. Illustrate using a Venn diagram. 3 INTRODUCTIONTODIRECTANDINDIRECTPROOFS
3.1
CONSTRUCT A DIRECT PROOF OF A PROPOSITION.
28. Show that if 𝑛 is an odd integer, then 𝑛2 + 1 is even.
29. Show that the product of two even integers is even.
30. Prove that if 𝑥 is rational and 𝑥 ≠ 0, then 1/𝑥 is rational.
31. Provide a specific counterexample to disprove the following false proposition.
“If 𝑎, 𝑏 ∈ Z, then 𝑎𝑏 ∈ Z.” CONSTRUCT AN INDIRECT PROOF OF AN IMPLICATION BY CONTRAPOSITIVE.
32. Show that if 𝑛 is an integer and 𝑛2 + 1 is even, then 𝑛 is odd. 33. Show that if 𝑥 + 5 is irrational, then 𝑥 is irrational.
CONSTRUCT AN INDIRECT PROOF OF A PROPOSITION BY CONTRADICTION.
34. Show that if 𝑝 is prime, then √𝑝 is irrational. You may assume without proof that if 𝑝|𝑛2 where
𝑛 ∈ Z, then 𝑝|𝑛.
35. Show that if 𝑎1, 𝑎2, 𝑎3 ∈ R, then at least one of these is greater than or equal to the average of
these numbers. Use a proof by contradiction.
CONSTRUCT EXISTENCE AND UNIQUENESS PROOFS.
36. Use an existence and uniqueness proof to show that the equation 15 − 2𝑥 = 3 has a unique
solution.
37. Prove the following proposition using an existence and uniqueness proof.
∀𝑟 ∈ Q+(∃!𝑠 ∈ Q+(𝑟𝑠 = 1))
3.2
3.3
3.4
Source: Rosen (2019). Discrete Mathematics and Its Applications, 8th Ed. McGraw-Hill.
Midterm Practice Problems
MTH 231
4 INTRODUCTIONTOMATHEMATICALINDUCTION
4.1
CONSTRUCT A PROOF USING THE WEAK FORM OF THE PRINCIPLE OF MATHEMATICAL INDUCTION
4.2
41. Show that 21 divides 4𝑛+1 + 52𝑛−1 for all positive integers 𝑛.
CONSTRUCT A PROOF USING THE STRONG FORM OF THE PRINCIPLE OF MATHEMATICAL
INDUCTION
42. Show that if 𝑛 ≥ 6 is an integer, then 𝑛 = 3𝑥 + 4𝑦 for some 𝑥, 𝑦 ∈ N. (Hint: consider previous
problems we’ve discussed about forming postage with two types of stamps.)
43. Find the flaw with the following “proof” of the false proposition “∀𝑎 ∈ R+(∀𝑛 ∈ N(𝑎𝑛 = 1)).” “Proof.” Let 𝑎 ∈ R+ be arbitrary. We proceed by strong induction on 𝑛.
(Base case.) When 𝑛 = 0, we have that 𝑎0 = 1 as needed.
(Inductive step.) Let 𝑘 ∈ N be arbitrary and assume that for all natural numbers 1 ≤ 𝑗 ≤ 𝑘, 𝑎𝑗 = 1. We then have that
𝑘+1 𝑎𝑘 · 𝑎𝑘 1 · 1
𝑎 =𝑎𝑘−1=1=1
so 𝑎𝑘+1 is also equal to 1 as needed.
44. Consider the sequence of real numbers with the following recursive definition:
𝑎𝑛 =3𝑎𝑛−1 +4𝑎𝑛−2, 𝑎1 =−1, 𝑎2 =21
a. Use the recursive definition to find the next two terms in the sequence 𝑎3 and 𝑎4.
b. Prove that the sequence can be defined with the explicit formula 𝑎𝑛 = 4𝑛 + 5(−1)𝑛.
38. Use mathematical induction to show that if 𝑎, 𝑏 ∈ R+, then (𝑎 + 𝑏)𝑛 ≥ 𝑎𝑛 + 𝑏𝑛. 39. Show that for all 𝑛 ∈ Z+, ∑𝑛 𝑖3 = (𝑛(𝑛+1))2.
𝑖=1
40. Prove that 6|(𝑛3 − 𝑛) whenever 𝑛 ∈ N.
2
Source: Rosen (2019). Discrete Mathematics and Its Applications, 8th Ed. McGraw-Hill.