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DISCRETE MATH: FINAL REVIEW
DR. DANIEL FREEMAN
1. Chapter 1 review
1) a. Does 3 = {3}?b. Is 3 ∈ {3}?c. Is 3 ⊆ {3}?c. Is {3} ⊆ {3}?c. Is {3} ∈ {3}?d. Does {3} = {3, 3, 3, 3}?e. Is {x ∈ Z|x > 0} ⊆ {x ∈ R|x > 0}?
2. Chapter 2 review
1) Construct a truth table for (∼ p ∧ q) ∨ (p∧ ∼ q).
2) Construct a truth table to show that ∼ (p ∧ q) is logically equivalent to ∼ p∨ ∼ q.What is the name for this law?
1
2 DR. DANIEL FREEMAN
3) Construct a truth table to determine if (p ∨ q)→ r is logically equivalent to (p→r) ∨ (q → r).
4) Use a truth table to determine if the following argument is logically valid. Write asentence which justifies your conclusion.
p→ q ∨ r
r → p ∨ q
∴ p→ r
DISCRETE MATH: FINAL REVIEW 3
You will be provided with the following information on the test.
2.1. Modus Ponens and Modus Tollens.• The modus ponens argument form has the following form:
If p then q.p
∴ q.• Modus tollens has the following form:
If p then q.∼ q∴∼ p.
2.2. Additional Valid Argument Forms: Rules of Inference.• A rule of inference is a form of argument that is valid. Modus ponens and
modus tollens are both rules of inference. Here are some more...Generalization p Elimination p ∨ q
∴ p ∨ q ∼ qSpecialization p ∧ q ∴ p
∴ p Transitivity p→ qProof by Division into Cases p ∨ q q → r
p→ q ∴ p→ rq → r Conjunction p∴ r q
Contradiction Rule ∼ p→ c ∴ p ∧ q∴ p
4 DR. DANIEL FREEMAN
5) Write a logical argument which determines what I ate for dinner. Number eachstep in your argument and cite which rule you use for each step.
a. I did not have a coupon for buns or I did not have a coupon for hamburger.b. I had hamburgers or chicken for dinner.c. If I had hamburgers for dinner then I bought buns.d. If I did not have a coupon for buns then I did not buy buns.e. If I did not have a coupon for hamburger then I did not buy buns.
DISCRETE MATH: FINAL REVIEW 5
3. Chapter 3 review
1) a. Give an example of a universal conditional statement.
b. Write the contrapositive of the example.
c. Write the negation of the example.
2) Write the following statements symbolically using ∀,∃,∨,∧,→. Then write theirnegation.
a. If x, y ∈ R then xy + 1 ∈ R.
b. Every nonzero real number x has a multiplicative inverse y.
c. Being divisible by 8 is not a necessary condition for an integer to be divisible by 4.
d. If I studied hard then I will pass the test.
6 DR. DANIEL FREEMAN
2) Is it true or false that every real number bigger than 4 and less than 3 must benegative? Explain why.
4. Chapter 4 review
1) Prove using the definition of odd: For all integers n, if n is odd then (−1)n = −1.
2) Prove using the definition of even: The product of any two even integers is divisibleby 4.
DISCRETE MATH: FINAL REVIEW 7
3) Prove: For each integer n with 1 ≤ n ≤ 5, n2 − n + 11 is prime.
4) Show that: .123123123... is a rational number.
5) Prove using the definition of divides: For all integers a, b, and c, if a divides b anda divides c then a divides b− c.
8 DR. DANIEL FREEMAN
6) Evaluate 60 div 8 and 60 mod 8.
7) Prove using the definition of mod: For every integer p, if pmod10 = 8 thenpmod5 = 3.
DISCRETE MATH: FINAL REVIEW 9
9 a) How do you prove a statement by contradiction?
9 b) Prove: There is no greatest integer. (for a more interesting problem, prove thatthere is no greatest prime number.)
9 a) How do you prove “∀x ∈ D, if P (x) then Q(x)” by contraposition?
9 b) Prove: For all integers a, b, and c, if a 6 |bc then a 6 |b.
10 DR. DANIEL FREEMAN
5. Chapter 5 review
1) a. Compute:∑5
i=2 i2
b. Compute:∏−1
i=−5 i
c. Compute:∑100
i=1(i)2 − (i− 1)2
d. Compute:∏50
i=2i(i−1)
(i+1)(i+2)
DISCRETE MATH: FINAL REVIEW 11
3 a) What are the steps for proving “For all integers n such that n ≥ a, P (n) is true”using mathematical induction?
3 b) Prove: For all integers n ≥ 1,n∑
i=1
5i− 4 =n(5n− 3)
2.
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6 a) What are the steps for proving “For all integers n such that n ≥ a, P (n) is true”using strong mathematical induction?
6 b) Prove: If s0 = 12, s1 = 29, and
sk = 5sk−1 − 6sk−2 ∀k ≥ 2,
then sn = 5 · 3n + 7 · 2n for all integers n ≥ 0.
DISCRETE MATH: FINAL REVIEW 13
6. Chapter 6 review
1) Let B = {n ∈ Z |n = 21r + 10 for some r ∈ Z} andC = {m ∈ Z |m = 7s + 3 for some s ∈ Z}. Prove that B ⊆ C.
2) Let A = {a, b, c, d, e}, B = {d, e, f, g} and C = {b, c, d, f}. What is (A−B) ∩ C?
14 DR. DANIEL FREEMAN
(1) Commutative Laws: For all sets A and B,
A ∪B = B ∪A and A ∩B = B ∩A.
(2) Associative Laws: For all sets A, B, and C,
(A ∪B) ∪ C = A ∪ (B ∪ C) and (A ∩B) ∩ C = A ∩ (B ∩ C).
(3) Distributive Laws: For all sets A, B, and C,
A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C).
(4) Identity Laws: For all sets A,
A ∪ ∅ = A and A ∩ ∅ = ∅.
(5) Complement Laws: For all sets A,
A ∪Ac = U and A ∩Ac = ∅.
(6) Double Complement Law: For all sets A,
(Ac)c = A.
(7) Idempotent Laws: For all sets A,
A ∪A = A and A ∩A = A.
(8) Universal Bound Laws: For all sets A,
A ∪ U = U and A ∩ ∅ = ∅.
(9) De Morgan’s Laws: For all sets A and B,
(A ∪B)c = Ac ∩Bc and (A ∩B)c = Ac ∪Bc.
(10) Absorption Laws: For all sets A and B,
A ∪ (A ∩B) = A and A ∩ (A ∪B) = A.
(11) Complements of U and ∅:U c = ∅ and ∅c = U
(12) Set Difference Law: For all sets A and B,
A−B = A ∩Bc
DISCRETE MATH: FINAL REVIEW 17
5) Prove or give a counterexample to the statement: For all sets A,B, and C,
A× (B ∩ C) = (A×B) ∩ (A× C)
18 DR. DANIEL FREEMAN
6) Prove or give a counterexample to the statement: For all sets A,B, and C,
A ∩ (B ∪ C) = (A ∪B) ∩ (A ∪ C)
DISCRETE MATH: FINAL REVIEW 19
7) Prove using the given set theory laws that for all sets A,B, and C,
A ∩ (B − C) = (A ∩B)− (A ∩ C)
20 DR. DANIEL FREEMAN
7. Chapter 7 review
9) Prove or give a counterexample to the statement: For all functions f : X → Y , andall sets C,D ⊆ Y ,
f−1(C ∪D) = f−1(C) ∪ f−1(D)
DISCRETE MATH: FINAL REVIEW 21
10) Prove or give a counterexample to the statement: For all functions f : X → Y , andall sets C ⊆ Y
f(f−1(C) = C
11) Prove or give a counterexample to the statement: For all functions f : X → Y , andall sets A ⊆ X
f−1(f(A)) = A
22 DR. DANIEL FREEMAN
8. Chapter 9 review
12) Prove that the number of r permutations of a set of n elements is
P (n, r) =n!
(n− r)!.
13) Prove that if X is a set of n elements then the number of subsets of X with relements is (
n
r
)=
n!
r!(n− r)!
DISCRETE MATH: FINAL REVIEW 23
14) Prove that if X is a set of n elements then the number of subsets of X is
N(P(X)) = 2n
15) Use problem 13) and 14) to prove thatn∑
r=0
n!
r!(n− r)!= 2n
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14) How many 3 digit numbers contain the digit 1? Justify your answer.
15) How many 3 digit numbers contain the digit 0? Justify your answer.
16) How many 3 digit numbers contain the digit 0 or the digit 1?
DISCRETE MATH: FINAL REVIEW 25
17) In a standard 52 card deck, there are 4 suits of 13 cards each.a. How many cards do you need to draw to be guaranteed of drawing at least 2 of the
same suit?
b. How many cards do you need to draw to be guaranteed of drawing at least 3 of thesame suit?
18) A group of 15 workers are supervised by 5 managers. Each worker is assignedexactly one manager, and no manager supervises more than 4 workers. Show thatat least 3 managers supervise 3 or more workers.
26 DR. DANIEL FREEMAN
19) How many ways are there to choose 5 people from a group of 14 to work as a team?
20) Suppose that there are 6 men and 8 women in the group. How many ways are thereto choose a team consisting of 2 men and 3 women.
21) Suppose that there are 6 men and 8 women in the group. How many ways are thereto choose a team consisting of 5 people such that at least one is a man.