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MATH-1401/3-001 and 002
Assignment 1
Due: Sept 20 and 21, 2011
beginning of class
1. Construct a truth-table for ( P Q) R.
2. Use rules of logical equivalences to prove each of the following:
(a) (P Q
MATH-1401/3-001 and 002
Assignment 3
Due: 0ct 18 and 19, 2011
beginning of class
Clearly justify all your steps. Recall that P(A) denotes the power set of a set A.
1. Suppose A = cfw_, 1, 2, cfw_1, 2,
MATH-1401/3-001 and 002
Assignment 3 Solutions
Fall 2011
beginning of class
Clearly justify all your steps. Recall that P(A) denotes the power set of a set A.
1. Suppose A = cfw_, 1, 2, cfw_1, 2, cfw_
MATH-1401/3-001 and 002
Assignment 2
Due: Oct 4 and 5, 2011
beginning of class
Special Instructions: For each question that requires you to prove a statement indicate
which method of proof you are usi
MATH-1401/3-001 and 002
Review
Instructor: O.Oellermann and N. Rampersad
Test 1
Fall, 2011
You will be tested on logic; proof methods (direct, contrapositive, and contradiction);
proving and disprovin
MATH-1401/3
Logical Equivalences
In the following T stands for true and F for false and P, Q, and R stand for statements.
1. Idempotence
(a) P P P
(b) P P P
2. Commutativity
(a) P Q Q P
(b) P Q Q P
3.
MATH-1401/3-001 and 002
Assignment 4
Due: Nov 1 and 2, 2011
beginning of class
Clearly justify all your steps.
1. Use induction to prove that 1 2 + 2 3 + 3 4 + . . . + n(n + 1) =
n(n+1)(n+2)
.
3
2. Sh
University of Winnipeg
MATH 1401-003 Discrete Math
Winter 2017
Assignment 3
Due 1 March 2017. At the end of this assignment there are some additional problems from
the textbook that you can try for ex
MATH-1401/3-001 and 002
Assignment 2 Solutions
Fall 2011
1. Let n Z. Prove that if n is odd, then n2 n is even.
We give a direct proof. Suppose that n is an odd integer. Then n = 2k + 1 for some
integ
University of Winnipeg
MATH 1401-003 Discrete Math
Winter 2017
Assignment 2
Due 3 Feb 2017. At the end of this assignment there are some additional problems from the
textbook that you can try for extr
Discrete Math Page 38
MATH-1401
Quantied Statements
Recall that if P (x) is an open sentence over domain S, then P (x) is a statement for
each x S. Another way to convert an open sentence into a state
MATH-1401
Discrete Math Page 53
Proof by Cases
Sometimes we divide a proof into parts called cases, and sometimes we further divide
the proof of a case into subcases.
Example 1: If we want to prove a
Discrete Math Page 45
MATH-1401
Chapter 3 - Methods of Proof
Mathematical Statements
A true mathematical statement whose truth is accepted without proof is called an
axiom. For example, the Well Order
Discrete Math Page 30
MATH-1401
Cartesian Products of Sets
Sets are unordered collections of objects.
cfw_x, y = cfw_y, x
An ordered pair, denoted by (x, y), consists of a pair of elements in which x
Discrete Math Page 22
MATH-1401
Set Operations and their Properties
There are several ways to combine two sets to produce another set.
The union of two sets A and B, denoted by A B, is the set of all
MATH-1401
Discrete Math Page 16
Chapter 2 - Sets and Subsets
A set is a collection of objects. The objects that make up a set are called elements (or
members). It is customary to use capital letters A
MATH-1401
Discrete Math Page 75
Sequences
A sequence is an ordered list of elements of some set. It may be nite or innite.
We denote an innite sequence a1 , a2 , a3 , . . . by cfw_an nN or simply cfw_
MATH-1401
Discrete Math Page 9
Tautologies and Contradictions
A compound statement is a tautology (contradiction) if it is true (false) for every
combination of truth values of its component statement
Discrete Math Page 67
MATH-1401
Chapter 4 - Mathematical Induction
The Principle of Mathematical Induction:
Let P (n) be an open sentence over domain N. If
(1) P (1) is true, and
(2) k N, P (k) = P (k
MATH-1401
Discrete Math Page 81
The Strong Principle of Mathematical Induction
Let S be a set of consecutive integers with least element m. Let P (n) be an open sentence
over domain S. If
(1) P (m) is
Discrete Math Page 60
MATH-1401
Proof by Contradiction
Suppose we want to prove a statement R. (R could be a quantied statement like
x S, P (x) = Q(x), or any other type of statement.) If we assume th