- 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 6= j.
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 6∈ B.
Since x ∈ A − B we know that x ∈ A and x 6∈ B, by the definition of set difference.
Thus x 6∈ B (Conjunctive Simplification). Note Any 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 6∈ Z+ . So R0 ∪ R1 6= Z+ .
2
Power Sets
Definition 6 The power set of a set A, denoted P(A), is the set of all subsets of A.
i.e. For any sets A and X
X ∈ P(A) ⇔ X ⊆ A
Example 7
1. Let A = {0, 1}
P(A) = {φ, {0}, {1}, {0, 1}}
2. Let A = {a, b, c}
P(A) = {φ, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Notes
1. X ∈ P(A) if and only if X ⊆ A.
2. For any set A, φ ∈ P(A) and A ∈ P(A).
Theorem 8 For any sets A and B, if A ⊆ B then P(A) ⊆ P(B).
To Prove: ∀ sets A, B, A ⊆ B → P(A) ⊆ P(B)
Proof:
Let A and B be sets with A ⊆ B.
We must show that every element of P(A) is in P(B).
Let X ∈ P(A). Thus X ⊆ A, by the definition of P(A).
Thus X ⊆ A ⊆ B, so X ⊆ B (5.2.1 3).
But if X ⊆ B then X ∈ P(B). Theorem 9 If a set A has n elements then P(A) has 2n elements.
S.W.P. (See p. 265)
2
- 5.1
3
Partitions, Power Sets and Cartesion Products
P. Danziger
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}.
3. Rn = R
| ×R×
{z. . . × R} = {(x1 , x2 , . . . , xn ) | x1 , x2 , . . . , xn ∈ R}.
n times
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,
4
yx = 00101.