Sets: Part 1

1
Sets Part 1

• Section 5.1

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Basic Definitions

• Set
• An unordered collection of objects, usually
  denoted by capital letter S
• Member, element
• Object in a set, usually denoted by lower case
  letter S a, b, c
• Set Membership
• a ? A denotes that a is an element of set A
• Cardinality
• Number of elements in a set, denoted S

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Remarks

• Ordering does not matter
• 1, 2, 3 2, 3, 1 1, 3, 2
• Repetitions are ignored
• 1, 1, 2, 3 1, 2, 3
• Elements in the set need not be of the same type
• 1, apple, ? is a set
• A set can contain other elements
• 1, 2, apple, ?, ?, 3 is a set with
  3 elements
• 1, 2
• Apple
• ?, ?, 3
• A set can be finite or infinite

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Predefined Sets

• Q - Set of all rational numbers
• R - Set of all real numbers (rational and
  irrational)
• Z - Set of all integers -3, -2, -1, 0, 1, 2, 3,
  
• N - Set of natural numbers 0, 1, 2, 3, 4,
• ? or - empty set
• U - Universal set, containing all elements under
  consideration
• Note
• Superscript to Z, Q, R indicates positive
  number gt 0
• Superscript - to Z, Q, R indicates negative
  number lt 0
• Superscript nonneg to Z, Q, R indicates positive
  number including 0
• Z 1, 2, 3, Z- -3, -2, -1 Znonneg
  0, 1, 2, 3, ..

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Defining Sets

• A set may be defined directly by listing every
  element
• S 2, 4, 6, 8, 10
• Or it may be defined indirectly by defining it in
  terms of other sets
• S x x ? Z, 1 lt x 10
• S x ? Z, 1 lt x 10
• In general
• S element element ? another set, list of
  conditions
• S element ? another set, list of conditions

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Set Theory and Predicate Calculus

• Think of set theory as predicate calculus
• Objects all the sets, all the numbers, etc
• Connectives ?, ?, -, x
• Predicates ?, ?, ?,
• So what can be written in predicate calculus
• ?A, ?B, Equal ( Union (A, B), Union (B, A) )
• can also be written in set notation.
• A ? B B ? A

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Predicates ?, ?, ?,

• S ? T S is a subset of T
• Every element of S is in T
• ?x, x ? S ? x ? T
• S T S equals T
• Exactly same elements in S and T
• (S ? T) ? (T ? S) Important for proofs!
• S ? T S is a proper subset of T
• S is a subset of T but S ? T
• (S ? T) ? (S ? T)
• X ? S x is an element of S
• Set membership

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Examples

• Membership
• 1 ? 1, 2, 3
• 1 ? 1, 2, 4, 5
• 1, 2 ? 1, 2, 4, 5
• 1, 2 ? 1, 2, 3, 4, 5
• Subset
• 1, 2 ? 1, 2, 3
• 1, 2 ? Z
• 1, 2 !? 1, 2
• Proper Subset
• 1, 2 ? 1, 2
• 1, 2 ? 1, 2, 3
• Z ? Z

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Examples (cont)

• Power Set P(S)
• Set of all subsets of S
• Cardinality of the power set is 2n where n is S
• If S 3, then P(S) 8

• ? ? S, ? sets S
• All subsets of Sa,b,c
• ?
• a,b,c
• a,b, b,c, a,c
• a,b,c
• What set has the same quantity as both an element
  and a subset?
• a, a

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Connectives ?, ?, -, c

• Union A ? B
• ?x, x? (A ? B) iff x?A ? x?B
• Like inclusive or, can be in A or B or both
• Complement Bc or B
• ?x, x? B iff x?U ? x?B
• Everything in the Universal set that is not in B

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More Set Operations

• Intersection A ? B
• ?x, x? (A ? B) iff x?A ? x?B
• A and B are disjoint if A ? B ?
• Difference A - B
• ?x, x? (A - B) iff x?A ? x?B

A
B
U
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Examples

• Let A n2 n?Z ? n?4 1, 4, 9, 16
• and B n4 n?Z ? n?4 1, 16, 81, 256
• A?B 1, 4, 9, 16, 81, 256
• A?B 1, 16
• A-B 4, 9
• B-A 81, 256

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Examples (cont)

• Let P x, t, m, s, Q t, s, y, R y, x
• U x, t, m, s, y
• (P ? R) ? Q x, t, s, y
• (R Q) ? P x, t, m, s
• R ? Q t, s
• (P ? Q)

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Cartesian Products

• Ordered n-tuple that takes both order and
  multiplicity into account
• Two ordered n-tuples (x1, x2, , xn) and (y1, y2,
  , yn) are equal iff x1 y1, x2 y2, xn yn
• Given two sets A and B, AxB set of all ordered
  pairs (a, b) where a ? A and b ? B
• A x, y
• B 1, 2, 3
• C a, b
• AxB (x,1), (x,2), (x,3), (y,1), (y,2), (y,3)
• BxC (1,a), (1,b), (2,a), (2,b), (3,a), (3,b)