Partitions, Power Sets and Cartesion Products P

- 5.1

Partitions, Power Sets and Cartesion Products P. Danziger

1 Set Partitions

Deﬁnition 1 Two sets are called disjoint if and only if they have no elements in common. • Formaly A and B are disjoint A B = φ. ⇔ ∩

A collection of sets A1,A2,...,An are mutually disjoint or pairwise disjoint if and only if • every pair of sets disjoint. More precisely, for all i, j = 1, . . . , n,

Ai Aj = φ whenever i = j. ∩ 6 Example 2

1. 1, 2 and 3,4 are mutually disjoint. { } { } 2. 1, 2, 3 and 2,3,4 are not mutually disjoint, since they both contain 2 and 3. { } { } + 3. Z and Z− are mutually disjoint 4. The intervals ( , 4], [ 4, 0) are not mutually disjoint since they contain the common element 4. −∞ − − − 5. The intervals ( , 4], ( 4, 0), [0, 2], [4, 10) are pairwise disjoint. −∞ − − Theorem 3 For any sets A and B, A B and B are mutually disjoint. − Proof: Let x be an element of A B, we must show that x B. Since x A B we know− that x A and x B, by6∈ the definition of set difference. ∈ − ∈ 6∈ Thus x B (Conjunctive Simplification). Note Any6∈ set A is mutually disjoint from its complement, Ac, since A Ac = φ. ∩

Deﬁnition 4 A collection of nonempty sets A1,A2,...,An is a partition of a set A if { }

1. A = A1 A2 ... An; ∪ ∪ ∪

2. A1,A2,...,An are mutually disjoint.

1 - 5.1 Partitions, Power Sets and Cartesion Products P. Danziger

Example 5

1. Let A = 0, 1, 2, 3, 4, 5 ,A1 = 0, 1 ,A2 = 2, 3 ,A3 = 4, 5 . Is A1,A2,A3 a partition of A? { } { } { } { } { }

Yes. Since A = A1 A2 A3 and A1,A2 and A3 have no common elements. ∪ ∪ 2. Consider the sets: R0 = x Z x = 2k for some integer k (even numbers). { ∈ | } R1 = x Z x = 2k + 1 for some integer k (odd numbers). { ∈ | } Is R0,R1 a partition of Z? { } Yes. Since every number is either odd or even R0 R1 = Z. ∪ No number is both odd and even, so R0 R1 = φ. ∩ + 3. Are the sets R0 and R1 above a partition of Z ? + + No. 2 R0, so 2 R0 R1, but 2 Z . So R0 R1 = Z . − ∈ − ∈ ∪ − 6∈ ∪ 6 2 Power Sets

Deﬁnition 6 The power set of a set A, denoted (A), is the set of all subsets of A. i.e. For any sets A and X P X (A) X A ∈ P ⇔ ⊆ Example 7 1. Let A = 0, 1 { } (A) = φ, 0 , 1 , 0, 1 P { { } { } { }} 2. Let A = a, b, c { } (A) = φ, a , b , c , a, b , a, c , b, c , a, b, c P { { } { } { } { } { } { } { }} Notes 1. X (A) if and only if X A. ∈ P ⊆ 2. For any set A, φ (A) and A (A). ∈ P ∈ P Theorem 8 For any sets A and B, if A B then (A) (B). ⊆ P ⊆ P To Prove: sets A, B, A B (A) (B) Proof: ∀ ⊆ → P ⊆ P Let A and B be sets with A B. We must show that every element⊆ of (A) is in (B). Let X (A). Thus X A, by the deﬁnitionP ofP (A). Thus X∈ P A B, so X ⊆ B (5.2.1 3). P But if X⊆ B⊆then X ⊆(B). ⊆ ∈ P Theorem 9 If a set A has n elements then (A) has 2n elements. P S.W.P. (See p. 265)

2 - 5.1 Partitions, Power Sets and Cartesion Products P. Danziger

3 Cartesian Products

+ Deﬁnition 10 Let n Z , and let x1, x2, . . . , xn be n (not necessarily distinct) elements of some ∈ set. The ordered n-tuple (x1, x2, . . . , xn) consists of x1, x2, . . . , xn together with the ordering. An ordered 2-tuple (x1, x2) is called an ordered pair. •

An ordered 3-tuple (x1, x2, x3) is called an ordered triple. •

Two ordered n-tuples (x1, x2, . . . , xn) and (y1, y2, . . . , yn) are equal if and only if •

x1 = y1 x2 = y2 ... xn = yn ∧ ∧ ∧ Thus (a, b) = (c, d) iﬀ a = c and b = d.

Deﬁnition 11 1. Given 2 sets A and B the Cartesian product of A and B, denoted A B (A cross B) is the set of ordered pairs (a, b) with a A and b B. × ∈ ∈ i.e. A B = (a, b) a A b B . × { | ∈ ∧ ∈ }

2. Given sets A1,A2,...,An the Cartesian product A1 A2 ... An is the set of all ordered × × × n-tuples (a1, a2, . . . , an).

i.e. A1 A2 ... An = (a1, a2, . . . , an) a1 A1 a2 A2 ... an An . × × × { | ∈ ∧ ∈ ∧ ∧ ∈ } Example 12

1. A = 1, 2 , B = 3, 4, 5 , A B = (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5) { } { } × { } 2. R R = R2 = (x, y) x, y R . × { | ∈ } n 3. R = R R ... R = (x1, x2, . . . , xn) x1, x2, . . . , xn R . × × × { | ∈ } | n times{z } 4. R N = (x, a) x R a N . × { | ∈ ∧ ∈ } 4 Alphabets and Strings

See p. 736 - 737.

Deﬁnition 13

1. An alphabet, Σ is a ﬁnite set. The elements of an alphabet are called symbols or characters.

Example 14

(a) ΣE = a, b, . . . , Y, Z - The standard alphabet for English. { } 3 - 5.1 Partitions, Power Sets and Cartesion Products P. Danziger

(b) ΣA = ASCII = ΣE !, @,..., ? - Standard alphabet for computer I/O. ∪ { } (c) Σ0 = 0, 1 - The natural alphabet of computers. { } 2. A string over an alphabet Σ is any ordered n-tuple of elements of Σ. We usually write strings with no commas or parantheses. We allow the empty string and denote it by the symbol . Generally, we use lowercase letters from the beginning of the alphabet a, b, c to denote single characters from an alphabet, and lowercase letters from the end of the alphabet u, v, w, x, y, z to denote strings of characters from an alphabet.

Example 15

(a) If Σ = Σ0 then , 0, 00, 01, 11, 01101100 are all strings over Σ.

(b) If Σ = ΣE then , “a”, “set”, “qwerty” are all strings over Σ.

3. The length of a string is the number of characters which make it up. The empty string always has length 0.

Example 16

(a) Σ = Σ0, 0 and 1 have length 1. 00, 01 and 11 have length 2. 01101100 has length 8.

(b) Σ = ΣE, “a” has length 1, “set” has length 3, “qwerty” has length 6.

4. Given an alphabet Σ Σn denotes the set of all strings of length n over Σ. Σ∗ denotes the set of all strings of any ﬁnite length (including 0) over Σ.

Example 17

Σ = Σ0. Σ0 = , { } Σ1 = Σ = 0, 1 , { } Σ2 = 00, 01, 10, 11 etc. { } 5. Given any two strings x and y over an alphabet Σ, the concatenation of x and y is the string xy.

Example 18 x = 01, y = 001, xy = 01001, yx = 00101.