Cardinality of a Set

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Cardinality of a Set

• The number of elements in a set.
• Let A be a set.
• If A ? (the empty set), then the
  cardinality of A is 0.
• b. If A has exactly n elements, n a
  natural number, then the cardinality of A is
  n. The set A is a finite set.
• c. Otherwise, A is an infinite set.

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Notation

• The cardinality of a set A is denoted by A
  .
• If A ? , then A 0.
• If A has exactly n elements, then A
  n.
• c. If A is an infinite set, then A ?.

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Examples A 2, 3, 5, 7, 11, 13, 17, 19 A
8 A N (natural numbers) N ? A
Q (rational numbers) Q ? A 2n n
is an integer A ? (the set of
even integers)
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DEFINITION Let A and B be sets. Then,
A B if and only if there is a one-to-one
correspondence between the elements of A
and the elements of B. Examples 1. A 1,
2, 3, 4, 5 B a, e, i, o, u 1? a,
2? e, 3? i, 4? o, 5? u B 5
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2. A N (the natural numbers) B 2n
n is a natural number (the even natural
numbers) n? 2n is a one-to one
correspondence between A and B.
Therefore, A B B ?. 3. A N
(the natural numbers) C 2n ?1 n is a
natural number (the odd natural
numbers) n? 2n ?1 is a one-to one
correspondence between A and C.
Therefore, A C C ?.
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Countable Sets
DEFINITIONS 1. A set S is finite if there
is a one-to-one correspondence between it and the
set 1, 2, 3, . . ., n for some natural
number n. 2. A set S is countably infinite
if there is a one-to-one correspondence between
it and the natural numbers N.
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• A set S is countable if it is either finite or
  countably infinite.
• A set S is uncountable if it is not countable.

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Examples
1. A 1, 2, 3, 4, 5, 6, 7, ? a, b,
c, d, . . . x, y, z are finite sets A
7, ? 26 . 2. N (the natural numbers), Z
(the integers), and Q (the rational numbers)
are countably infnite sets that is, Q
Z N.
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• 3. I (the irrational numbers) and ?
• ? (the real numbers) are uncountable sets
• that is
• I gt N and ? gt N.

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Some Facts

• A set S is finite if and only if for any proper
  subset A ? S, A lt S that is, proper
  subsets of a finite set have fewer elements.
• Suppose that A and B are infinite sets and A
  ? B. If B is countably infinite then A is
  countably infinite and A B.

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• 3. Every subset of a countable set is countable.
• If A and B are countable sets, then A ? B
• is a countable set.

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Irrational Numbers, Real Numbers
Irrational numbers points on the real line
that are not rational points decimals that
are neither repeating nor terminating. Real
numbers rationals ? irrationals
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is a real number
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is not a rational number, i.e.,
is an irrational number.
Proof Suppose is a rational number.
Then . . .
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Other examples of irrational numbers

• Square roots of rational numbers that are not
• perfect squares.
• Cube roots of rational numbers that are not
• perfect cubes.
• And so on.
• ? ? 3.14159, e ? 2.7182182845

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• Algebraic numbers
• roots of polynomials with integer coefficients.
• Transcendental numbers
• irrational numbers that are not algebraic.

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THEOREM The real numbers are uncountable! Proo
f Consider the real numbers on the interval
0,1. Suppose they are countable. Then . . .
Arrive at a contradiction. COROLLARY The
irrational numbers are uncountable. Proof Real
numbers rationals ? irrationals
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The Real Line
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Absolute Value
DEFINITION Let a be a real number. The
absolute value of a, denoted a, is given by
Geometric interpretation a is the distance
on the real number line from the point a to the
origin 0.
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55, ?33
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Absolute value inequalities

• Find the real numbers x that satisfy
• 1. x lt 3
• 2. x ? 2
• 3. x ? 3 ? 4
• 4. x 2 gt 5
• 5. 2x ? 3 lt 5

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Answers

• (?3,3) ?3 lt x lt 3
• (??,?2 ? 2,?) x ? ?2 or x ? 2
• ?1,7 ? 1 ? x ? 7
• 4. (??,?7) ? (3,?) x lt ?7 or x gt 3
• 5. (?1,4) ?1 lt x lt 4