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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
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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
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Chapter 3 - Methods of Proof
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
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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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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,
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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
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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 .
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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.
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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.
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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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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