Math 310
Solutions to Selected Problems from HW #6
2.1 #16 Find two sets such that A B and A B.
Solution:
First, we should note that to have any chance of A being both an element of B and a subset of B, the
set B that we choose must contain at least one s
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AnEuEerpath:c>b>d>c
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12.
a). b).
The least number of comparisons is n - 1. The least number of comparisons is 1.
c).
The least number of comparisons is 1.
22.
The list should be all check
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1.1 # 12
a. If you have the u, then you miss the nal examination.
b. You do not miss the nal examination, if and only if you pass the course.
c. If you miss the nal examination then you do not pass the course.
d. You have the u or you miss the nal examina
CONCORDIA UNIVERSITY
2.
DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING
Mathematics for Computer Science
COMP 232/2
i) It is clear that R is the graph of a function from A to B if and only if
both P1 and P2 hold, where
P1 a A b B [ (a, b) R ],
FALL
Problem (1.7.38)
Let Aj cfw_ -2, -1, 0, 1, , j. Find
=
a)
n
Aj
j=1
Each Aj is the set cfw_ j, so every Aj fully contains the sets Aj-1 Aj-2 etc. as subsets. Therefore, the union of the sets A1
through An is exactly An. We can take this one step further a
Problem (1.6.4)
Suppose that A=cfw_2, 4, 6, B=cfw_2, 6, C=cfw_4, and D=cfw_4, 6, Determine which of these sets are subsets of which other
6,
8.
of
these sets.
Every set is a subset of itself
B is a subset of A
C is a subset of both A and D.
Problem (1.6.1
1.4 # 8
a. For every animal, if it is a rabbit, then it hops. Equivalently, Every animal that is a rabbit
hops.
b. Every animal is a rabbit and hops.
c. There exists an animal such that if it is a rabbit then it hops.
d. There exists an animal which is a
CONCORDIA UNIVERSITY
6. Use the Euclidean algorithm to nd
DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING
COMP 232/2
Mathematics for Computer Science
a) gcd(123, 277),
FALL 2009
b) gcd(1529, 14039).
Show all your steps.
Assignment 3 : Sections Q, R,
CONCORDIA UNIVERSITY
6. Let A =
DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING
Mathematics for Computer Science
COMP 232/2
FALL 2009
2r + 3s r N, s N, r = s ,
B=
2r + 3s r N, s N, r = s ,
C=
n N 0 n 15 .
Find the following.
Assignment 2 : Sections
CONCORDIA UNIVERSITY
DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING
4. Let P (x, y) denote the statement x < y 3 + 1, where x, y Z. What is the
truth value of each of the following? Explain your answers.
Mathematics for Computer Science
FALL 2009
A
I am grateful to Bryan Gingras for supplying the following information
to our class:
Please make any corrections that you notice and forward to me.
1.1 #10 in 6th Edition corresponds to 1.1 #14 in 7th edition
1.1 #26 = 1.1 #30
1.1 #34 = 1.1 #38
1.1 #38 =
Problem Sheet 1
MATH 363
Winter 2017
On this sheet all but the last two exercises are assessed and solutions are due before 5pm on
Thursday 19 January 2017.
1. For each of the following, decide whether the compound proposition given is a tautology,
a cont
Problem Sheet 2
MATH 363
Winter 2017
On this sheet all exercises are assessed and solutions are due before 5pm on Thursday 2
February 2017.
1. Let a1 and a2 be strictly positive real numbers and define their geometric mean by =
a1 a2 . Prove that if a1 >
Section 1.1
Propositional Logic
1
CHAPTER 1
The Foundations: Logic and Proofs
SECTION 1.1
Propositional Logic
2. Propositions must have clearly dened truth values, so a proposition must be a declarative sentence with no
free variables.
a) This is not a pr
MATH 363 ASSIGNMENT 1
PUTRA MANGGALA
1. Determine whether ( p ( p q) q is a tautology.
Answer: We can write down the truth table and check if for all inputs we get T . But we wont, and
thus its not a tautology. A nicer proof is to reduce this into a simpl
MATH 363
Assignment 1
Due in class January 22
The assignment is worth 4% of your nal grade. Recall that the marking scheme has
been changed to max(20% Assignments + 20% Midterm + 60% Final, 20% Assignments +
80% Final).
Answer two questions out of questio
MATH 363 ASSIGNMENT 2
PUTRA MANGGALA
1. Prove: n > 6, 3n < n!.
Answer: We can do an induction on n. Take n = 7 as the base case, then 37 = 2187 < 5040 = 7!.
Hypothesing that this is true for n = k > 6, we have 3k+1 = 3(3k ) = 3(k !) < (k + 1)k ! = (k + 1)
MATH 363
Assignment 2
Due in class February 10
Assignment 2
This assignment is worth 4% of your nal grade. Recall that the marking scheme has been changed to
max(20% Assignments + 20% Midterm + 60% Final, 20% Assignments + 80% Final).
Answer all questions