MATH/EECS 1028: Discrete Math for Engineers
Winter 2015
Assignment 3 (Released April 06, 2015) Solutions
Question 1
[5 points] Find the coecient of x4 in the expansion of (1 + 3x + 2x3 )12 ?
Solution: Let us rewrite the given expression as (1 + 3x + 2x3 )
MATH/EECS 1028: Discrete Math for Engineers, Winter 2015
Quiz 5 (Apr 15, 2015) Solutions
Questions:
1. (1 point) Let Canadian(x), M adeOf f er(x), Qualif ied(x) be the statements x is a
Canadian citizen, x is given a job oer, x is qualied for the job resp
MATH/EECS 1028: Discrete Math for Engineers
Winter 2015
Solutions
Question 1
[4 points] Prove that log3 + log 3 > 2. Do not use a calculator, or a lookup table to compute numerical
values of the logarithms.
Solution: First we need to prove that log3 > 1.
MATH/EECS 1028 First test (version 1)
Winter 2015
Instructor: S. Datta
Solutions
1. (3 points) Construct a truth table for the implication (p q) (p q)
Solution: The truth table is given below.
p
T
T
F
F
q pq
T
T
F
T
T
T
F
F
pq
T
F
F
F
(p q) (p q)
T
F
F
T
MATH/EECS 1028 First test (version 2)
Winter 2015
Instructor: S. Datta
Solutions
1. (3 points) Construct a truth table for the implication (p q) (p q)
Solution: The truth table is given below.
p q
T T
T F
F T
F F
pq
T
F
F
F
p q
F
T
T
T
(p q) (p q)
F
T
T
T
MATH/EECS 1028 Second test (version 1)
Winter 2015
Feb 27, 2015
Solutions
1. (4 points) Use induction to prove that for n N, 1 + 4 + 7 + . . . + (3n 2) = n(3n 1)/2.
Solution: We prove this using induction on n.
Base Case: For n = 1 the left hand side is 1
MATH/EECS 1028 Second test (version 1)
Winter 2015
Feb 27, 2015
Solutions
1. (4 points) Use induction to prove that for n N, 1 + 3 + 5 + . . . + (2n 1) = n2 .
Solution: We prove this using induction on n.
Base Case: For n = 1 the left hand side is 1 and t
MATH/EECS 1028 Second test
Winter 2015
Sample questions solutions
1. Suppose A, B, C are sets. Prove or disprove: (A B) C = A (B C).
Solution: We can disprove this with a counterexample. Let A = cfw_1, 2, 3, 4, 5, B = cfw_2, 3, 5, 6 and C =
cfw_4, 5, 6, 7
MATH/EECS 1028: Discrete Math for Engineers, Winter 2015
Quiz 3 v1 (Mar 20, 2015) Solutions
Questions:
1. (1 point) Prove that given any 8 people at least two were born on the same day of the
week.
Solution: Since each of the 8 people are born on one of s
MATH/EECS 1028: Discrete Math for Engineers
Winter 2015
Tutorial 1 (Week of Jan 12, 2015)
Notes:
1. Assume R to denote the real numbers, Z to denote the set of integers (. . . , 2, 1, 0, 1, 2, . . .)
and N to denote the natural numbers (1, 2, 3, . . .).
2
MATH/EECS 1028 Second test (version 2)
Winter 2017 - Solutions Instructor: S. Datta
1. (6 points) Let g : Z Z Z Z be defined by g(x, y) = (2x, x + y). Is the function g
one-one (injective)? Is it onto (surjective)? Justify your conclusions.
Solution: The
MATH/EECS 1028 Second test (version 1)
Winter 2017
Mar 1, 2017
Instructor: S. Datta
1. (6 points) Let g : Z Z Z Z be defined by g(x, y) = (x + y, 3y). Is the function g
one-one (injective)? Is it onto (surjective)? Justify your conclusions.
Solution: The
MATH/EECS 1028 Winter 2017: First test (version 1) Solutions
Instructor: S. Datta
1. (3 points) Determine, by constructing a truth table, whether the following is a tautology.
(p (p q) q
Solution: The truth table is given below.
p
T
T
F
F
q pq
T
T
F
F
T
T
EECS 1520
Computer Use: Fundamentals
Practice questions on number systems
A. For the following 6-bit binary numbers, a) convert to decimal, b) convert to octal, c) convert to
hexadecimal
1.
2.
3.
4.
110101
001011
111111
011101
[Ans: a) 53 b) 65
[Ans: a) 1
MATH/EECS 1028: Discrete Math for Engineers, Winter 2015
Quiz 4 v2 (Apr 6, 2015) - Solutions
Questions:
1. (1 point) How many 6-nucleotide DNA sequences do not contain the sequence CTGG?
Note: DNA sequences are strings over cfw_A, C, T, G and the CTGG can
MATH/EECS 1028 First test
Winter 2015
Jan 30, 2015
Sample questions
1. Propositional Logic.
(a) Construct a truth table for the implication p q
Solution:
p q q p q
T T F
F
T F T
T
F T F
T
F F T
T
(b) Let p be the proposition You have the u, q be the propo
MATH/EECS 1028: Discrete Math for Engineers, Winter 2015
Quiz 4 v3 (Apr 10, 2015) Solutions
Questions:
1. (1 point) How many 7-nucleotide DNA sequences do not contain the sequence CTGG?
Note: DNA sequences are strings over cfw_A, C, T, G and the CTGF cann
EECS 1028 3.00 Discrete Mathematics for Engineers
(Cross-listed with MATH1028 3.00)
Introduction to abstraction; use and development of precise formulations of mathematical ideas, in particular as they
apply to engineering; introduction to propositional l
MATH/CSE 1019 Final Examination
Fall 2011
December 17, 2011
Instructor: S. Datta
1. (2 points) Evaluate the innite geometric series
9
9
9
+
+
+ .
10 100 1000
Note that the left hand side is 0.99 . . . Use the answer above to ll in the blank below
Solution
MATH/EECS 1028 Winter 2015
Third test (version 1) : Solutions
Instructor: S. Datta
1. (6 points) Let P (n) be the statement that a postage of n cents can be formed using just
3-cent and 5-cent stamps. Prove using strong induction that P (n) is true for n
MATH/EECS 1028: Discrete Math for Engineers, Winter 2015
Quiz 4 v1 (Apr 6, 2015) Solutions
Questions:
1. (1 point) How many 5-nucleotide DNA sequences do not contain the sequence CTG?
Note: DNA sequences are strings over cfw_A, C, T, G and the CTG cannot
MATH/EECS 1028: Discrete Math for Engineers
Winter 2015
Tutorial 7 (Week of Mar 2, 2015)
Notes:
1. Assume R to denote the real numbers, Z to denote the set of integers (. . . , 2, 1, 0, 1, 2, . . .)
and N to denote the natural numbers (1, 2, 3, . . .).
2.
MATH/EECS 1028 Winter 2015
Third test (version 2) - Solutions
Instructor: S. Datta
1. (6 points) Cardinality
(a) (2 points) We argued in class that the set of all possible Java programs is countable.
Write down one way to argue this.
Solution: Convert the
MATH/EECS 1028 Winter 2017: First test (version 2) Solutions
Instructor: S. Datta
1. (3 points) Construct a truth table for the implication (p q) (p q).
Solution: The truth table is given below.
p q
T T
T F
F T
F F
pq
T
F
F
F
p q
F
T
T
T
(p q) (p q)
F
T
T
Mathematical Induction
Very simple
Very powerful proof technique
Guess and verify strategy
EECS 1028, Winter 2015
69
Basic steps
Hypothesis: P(n) is true for all positive
integers n
Base case/basis step (starting value)
Inductive step
Formally:
[ P(
Assertions
Axioms
Proposition, Lemma, Theorem
Corollary
Conjecture
EECS 1028, Winter 2015
52
Types of Proofs
Direct proofs (including Proof by cases)
Proof by contraposition
Proof by contradiction
Proof by construction
Proof by Induction
Other techniques