DISCRETE MATH: FINAL REVIEWmath.slu.edu/~freeman/Discrete_Lecture_spring2012_27.pdf · DISCRETE...

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?

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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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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)

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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.

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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?

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(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

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3) Prove the second part of the Distributive Law

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4) Prove the second part of De Morgan’s Law.

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5) Prove or give a counterexample to the statement: For all sets A,B, and C,

A× (B ∩ C) = (A×B) ∩ (A× C)

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6) Prove or give a counterexample to the statement: For all sets A,B, and C,

A ∩ (B ∪ C) = (A ∪B) ∩ (A ∪ C)

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7) Prove using the given set theory laws that for all sets A,B, and C,

A ∩ (B − C) = (A ∩B)− (A ∩ C)

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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)

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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

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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)!

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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?

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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.

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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.