Problem assignments are from Ensley and Crawley: Discrete Mathematics: Mathematial Reasoning and Proof with Puzzles, Patterns, and Games, or are given in full here.
This is full text, copied from the book, for the first seven assignments. Questions for the remaining assignments will be on the main homework page.
(b) an = 5n - 2
(d) an = 3n + 1
(f) an = 3n2 + n.
(b) a1 = 2; an = 5an-1
(d) a1 = 4; an = 4 + an-1
(f) a1 = 6; an = 3 + an-1
(b) ∑3k=1 7k
(d) ∑4k=0 1/2k
(a) a1 = 1; an = an/2 + n if n is even;
an = a(n-1)/2 + n if n is odd.
(b) a1 = 1; an = 3an/2 + 2 if n is even;
an = 3a(n-1)/2 + 2 if n is odd.
(c) a1 = 1; an = 3an/2 + n if n is even;
an = 3a(n-1)/2 + n if n is odd.
(a) A is lying and B or C is truthful.
(b) A and B are lying, or A and C are truthful.
(c) At least two people are telling the truth.
(d) Exactly two people are telling the truth.
(b) ¬t ∧ ¬d
(d) (t ∧ d) ∨ (¬t ∧ d)
(a) Jillian likes playing in the sand or volleyball, but she does not like sailing.
(b) Jillian likes playing in the sand, and she likes sailing or volleyball.
(c) Jillian likes playing in the sand and volleyball, or she likes sailing.
(d) Jillian likes playing in the sand and sailing, or she likes volleyball and sailing.
(b) The staff is friendly, or else they are not friendly but very well paid.
(d) You play sports or you play mahjong or nobody knows your name.
(e) You do not play mahjong or you do not play sports, and nobody knows your name.
(b) (p ∨ q) ∧ (¬p ∨ q) and q
(d) p ∧ (q ∨ r) and (p ∧ q) ∨ r
(e) (p ∨ q) ∧ (q ∨ r) and (p ∧ r) ∨ q
(b) (p ∨ t) ∧ (p ∨ c) ≡ p [Remember that "t" is a tautology, also written "1", and "c" is a contradiction, also written "0".]
(c) q ∧ (p ∨ r) ≡ (p ∧ q) ∨ (q ∧ r)
(a) Both x and y are positive.
(b) At least one of x and y is positive. (Typo corrected 18 September.)
(c) Exactly one of x and y is positive.
(d) Neither x nor y is positive.
(a) If x = 8 and y = 3, which of (a)-(d) are true?
(b) If x = -5 and y = 0, which of (a)-(d) are true?
(c) If x = 7 and y = -7, which of (a)-(d) are true?
(d) If x = 0 and y = 0, which of (a)-(d) are true?
(e) If x = -3 and y = -10, which of (a)-(d) are true?
(f) If x = -8 and y = 2, which of (a)-(d) are true?
(a) For every integer , 2n ≠ 9.
(b) There exists a triangle T that is equilateral and has perimeter 10.
(c) Every circle has an integer diameter or an integer area.
(d) Every two real numbers have an integer in between.
(a) Even numbers are never prime.
(b) Triangles never have four sides.
(c) There are no integers a and b for which a2/b2 = 2.
(d) No square number immediately follows a prime number.
(a) ∀a∈R, ∀b∈Z, a2 + b2 ∈ Z.
(b) ∃y∈ R, ∀x∈R, x + y = x.
(c) ∀x∈Z, ∃y∈R, x = 27.
(d) ∀x∈Z, ∃y∈R, x/y = 2.
(a) Every time you roll a "6", you have to take a card.
(b) There is a day in your life that is better than every other day.
(c) In every good book, there is a plot twist or a surprise ending.
(d) Every math course has a topic that everyone finds easy to do.
(a) For all odd integers b, there is no real number x such that x2 + bx + 15 = 0.
(b) For every two real numbers x and y, there is an integer n such that x < n < y.
(c) For every pair of integers that sum to 5, at least one of the numbers must be bigger than 2.
(b) If you don't attend the concert, you will get an F for the course.
(d) If you don't eat your breakfast, you will be hungry.
(f) If a quadrilateral is a square, it has four equal sides and four equal angles.
(g) If a triangle has either two equal sides or two equal angles, then it is an isosceles triangle.
(b) ¬p → (p ∨ q)
(d) ¬p ∧ (¬q → p)
(f) (p ∧ q) → (p ∨ r)
(a) If Alaina likes basketball, then she likes swimming and gymnastics.
(b) If Alaina likes gymnastics, then she likes swimming and basketball.
(c) If Alaina dislikes gymnastics or dislikes swimming, then she dislikes basketball.
(d) Alaina dislikes basketball or she likes both swimming and gymnastics.
(b) If sin(x) = 0, then cos(x) = 1 (type of x = "real number")
(d) If x3 = x, then x2 = 1 (type of x = "real number")
(b) Some computer science majors take discrete mathematics.
(d) Not every computer science major takes discrete mathematics.
(f) You must take discrete mathematics if you are a computer science major.
(g) Some computer science majors do not take discrete mathematics.
(a) For each n > 1 and m > 1, if mn is a multiple of 4, then either m or n is a multiple of 4.
(b) For each n > 1 and m > 1, if mn is a multiple of 3, then either m or n is a multiple of 3.
(c) For each n > 1 and m > 1, if mn is a perfect square, then either m or n is a perfect square.
(d) For each n > 1 and m > 1, if m is even and n is odd, then either m + n2 or m2 + n is prime.
(b) If n is even, then n + 8 is even.
(d) If n is odd, then n + 1 is even.
(f) If n is divisible by 4, then n3 - n is divisible by 4.
(g) If n is divisible by 3, then n2 is divisible by 9.
(b) The product of an odd integer and an even integer is always even.
(a) If n is an even integer greater than 2, then 2n - 1 is not prime.
(b) Every odd perfect square can be written in the form 8k + 1. (Hint: Exercise 10c can help.)
(c) 7 is the only prime number that precedes a perfect cube? (Hint: How does the polynomial x3 - 1 factor?)
(d) Every even perfect square is a multiple of 4. (Hint: Use Theorem 7.)
Last modified 23 September 2016