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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 statement is to quantify
it. There are two types of quantier
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 statement P (n) is true for all integers n, we may
brea
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 Ordering Principle stated below is an axiom.
The Well Orderi
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 is the rst
coordinate and y is the second coordinate. T
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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 elements which
belong to A or B, or both. That is,
A B
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, B, C, S, T, X, Y, . . . to denote sets
and lower case
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_an , where the n-th
term of the sequence is an .
Exampl
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 statements.
Examples:
MATH-1401
Discrete Math Page 10
Logical Eq
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 + 1) is true,
then P (n) is true for all n N.
Proof (b
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 true, and
(2) (i cfw_m, m + 1, . . . , k, P (i) = P (k
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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 that R is
false, and then from this assumption, we are ab