DDM Chapter 3

bit

A character in a binary string.

Set

A collection of objects.

x; y

The first entry of the ordered pair (x, y) is _____ and the second entry is _____.

length

The number of characters in the string.

Positive

x, a number, x > 0.

False.

(T/F) AXB = BXA.

True.

(T/F) If two sets are equal then they have exactly the same elements, e.g. D = {3, 4, 5} and E = {5, 3, 4}, then D=E

True.

(T/F) The Cartesian product can be denoted more generally as: A^k=A×A×⋯×A

False.

(T/F) The number 0 can be positive or negative.

B, B, A, A, B, and A.

1) -3 ∈ Z+ A)True B) False 2) 0 ∈ Z+ A)True B) False 3) 0 is a non-negative integer. A)True B) False 4) 5 ∈ R+ A)True B) False 5) -5 is a member of the set of all non-negative integers. A)True B) False 6) 0 ∈ Q A)True B) False

Roster Notation

A list of the elements enclosed in curly braces with the individual elements separated by commas; e.g. A = {2, 4, 6, 10}, A = {10, 6, 4, 2}

partition

A non-empty set A is a collection of non-empty subsets of A such that each element of A is in exactly one of the subsets.

Negative

A number x, x < 0.

Non-negative

A number x, x ≥ 0.

ordered pair

A pair of items that is written (x, y).

string

A sequence of characters.

pairwise disjoint

A sequence of sets, A1, A2, ..., An, that every pair of distinct sets in the sequence is disjoint (i.e., Ai ∩ Aj = ∅ for any i and j in the range from 1 through n where i ≠ j)..

Set Builder Notation

A set defined by specifying that the set includes all elements in a larger set that also satisfy certain conditions; e.g. A = { x ∈ S : P(x) }

Proper Subset

A set in which A ⊆ B and there is an element of B that is not an element of A (i.e., A ≠ B); denoted as A ⊂ B.

Subset

A set in which every element in A is also an element of B; denoted as A ⊆ B.

Finite Set

A set that has a finite number of elements.

Infinite Set

A set that has an infinite number of elements.

n-bit string

A string of length n.

binary string

A string whose alphabet is {0, 1}.

Set Identity

An equation involving sets that is true regardless of the contents of the sets in the expression.

ordered triple

An ordered list of three items and is denoted (x, y, z). .

B, A, B, A, and A.

Consider the following set: A = { 2, 3, 5, 7, 14 } 1) 2 ∈ P(A). A) True B) False 2) { 2 } ∈ P(A). A) True B) False 3) { 2, 3 } ∈ A. A) True B) False 4) { 2, 3 } ∈ P(A). A) True B) False 5) |P(A)| = 32. A) True B) False

A, A, B, and B.

Consider the following set: A = { 3, 4, { 3, 4 }, { 1, 2, 3 }, 5 } 1) The cardinality of A is 5. A) True B) False 2) { 3 } ⊆ A. A) True B) False 3) { 1 } ⊆ A. A) True B) False 4) { 1, 2, 3 } ⊆ A. A) True B) False

A, B, B, A, B.

Consider the following sets: A = { 1, 2, 3 } B = { x, y } 1) (1, y) ∈ A x B A) True B) False 2) (1, y) ∈ B x A A) True B) False 3) A ⊆ A x B A) True B) False 4) (2, 3) ∈ Z x Z A) True B) False 5) |A x B| = 5 A) True B) False

Power set

Set, denoted as P(A), that is the set of all subsets of A

A, B, B, A.

Consider the following sets: A = { 1, 2, 3 } B = { x, y } C = { u, v, w } D = { +, * } 1) (w, y, 2) ∈ C x B x A A) True B) False 2) A x B x C ⊆ A x B x C x D Hint: The set A x B x C contains triples. The set A x B x C x D contains 4-tuples. A) True B) False 3) (1, x, u, +) ∈ B x A x C x D A) True B) False 4) (1, 2, +) ∈ A x A x D A) True B) False

B, A, A, B, A, A, A, and B.

Consider the following sets: A = { 3, 4, 5 } B = { 4, 5, 3 } C = { x ∈ Z: x is odd } D = { 3, 5, 7, 9 } 1) A ⊂ B? A) Yes B) No 2) A ⊆ B? A) Yes B) No 3) C ⊂ Z? A) Yes B) No 4) B ⊆ D? A) Yes B) No 5) D ⊆ C? A) Yes B) No 6) N ⊂ Z ⊂ Q ⊂ R ? A) Yes B) No 7) Is the following statement true? For any two sets, X and Y, if X ⊂ Y, then X ⊆ Y. A) Yes B) No 8) Is the following statement true? For any two sets, X and Y, if X ⊆ Y, then X ⊂ Y. A) Yes B) No

A, B, B, B, A, and A.

Consider the following sets: A = { 4, 6, 3 } B = { 2, 4, 6, ..., 20 } C = { 2, 4, 6, ... } D = { 3, 4, 6 } 1) A = D A) True B) False 2) 5 ∈ A A) True B) False 3) 5 ∈ ∅ A) True B) False 4) C is finite. A) True B) False 5) |B| = 10 A) True B) False 6) |A| = |D| A) True B) False

B, A, A.

Define the set A = { a, b, c } 1) (a, a, a, a, a) ∈ A^4 A) True B) False 2) (a, b, b, a) ∈ A^4 A) True B) False 3) (2, 3, 2) ∈ R^3 A) True B) False

0011

Define the string t = 001. What is t1?

001

Define the string t = 001. What is tλ?

1101001

Define the strings s = 1101 and t = 001. What is st?

Complement

Denoted (*Line on top of A*) A, is the set of all elements in U that are not elements of A. U-A.

Difference

Denoted A - B, is the set of elements that are in A but not in B.

Union

Denoted A ∪ B and read "A union B", is the set of all elements that are elements of A or B.

Symmetric Difference

Denoted A ⊕ B, is the set of elements that are a member of exactly one of A and B, but not both. ( A - B ) ∪ ( B - A )

Intersection

Denoted as A ∩ B and read "A intersect B", is the set of all elements that are elements of both A and B.

Universal set

Denoted by the variable U, is a set that contains all elements mentioned in a particular context.

*Z*

Denotes the set of all integers; e.g. ..., -2, -1, 0, 1, 2, ...

*N*

Denotes the set of natural numbers: All integers greater than or equal to 0; e.g. 0, 1, 2, ...

*Q*

Denotes the set of rational numbers: All real numbers that can be expressed as a/b, where a and b are integers and b ≠ 0; e.g. 0, 1/2, 5.23, -5/3

*R*

Denotes the set of real numbers; e.g. 0, 1/2, 5.23, -5/3, π, 2√

ordered n-tuple

For n ≥ 4, an ordered list of n items. E.g. (w, x, y, z)

Cartesian product

For two sets, A and B, denoted A x B, is the set of all ordered pairs in which the first entry is in A and the second entry is in B. A x B = { (a, b) : a ∈ A and b ∈ B }

2^n

How do you find the cardinality of a power set?

|A| * |B|

If A and B are finite sets, then |AXB| (cardinality of set A and B) = ________.

proper subset

If A ⊆ B and there is an element of B that is not an element of A (i.e., A ≠ B), then A is a _________ _________ of B, denoted as A ⊂ B.

subset; subset

If every element in A is also an element of B, then A is a _______ of B, denoted as A ⊆ B. If there is an element of A that is not an element of B, then A is not a _______ of B, denoted as A ⊈ B.

concatenation

If s and t are two strings, then the combination of s and t (denoted st) is a longer string obtained by putting s and t together.

:

In A = { x ∈ S : P(x) }, denotes "Such that."

()

Indicates the order of entries is significant.

Cardinality

Let A be a finite set, denoted by |A|, is the number of elements in A; e.g. A = {2, 4, 6, 10}, then |A| = 4

no

Let X = {x, y, z}. Is the string zzyzx an element in X^4?

Venn diagrams

Pictorially represent sets; a rectangle is used to denote universal set U and oval shapes are used to denote sets within U.

Elements

The objects in a set.

alphabet

The set of characters used in a set of strings.

Empty Set

The set with no elements and is denoted by the symbol ∅; also called a null set.

-

The superscript used to indicate the negative elements of a particular set; e.g. Z-.

+

The superscript used to indicate the positive elements of a particular set; e.g. R+.

A

The universe set is R. A1 = { x ∈ R: x ≤ -4 } A2 = { x ∈ R: -4 < x < 4 } A3 = { x ∈ R: x ≥ 4 } Which of the following is true? A) A1, A2, and A3 form a partition of R. B) A1, A2, and A3 are not mutually disjoint. C) A1 ∪ A2 ∪ A3 ≠ R.

B

The universe set is Z+. A1 = { x ∈ Z+: x is prime } A2 = { x ∈ Z+: x is odd and x is a composite number } A3 = { x ∈ Z+: x is even } A4 = { 1 } Which of the following is true? A) A1, A2, A3, and A4 form a partition of Z+. B) A1, A2, A3, and A4 are not mutually disjoint. C) A1 ∪ A2 ∪ A3 ∪ A4 ≠ Z+

A

The universe set is Z. O = { x ∈ Z: x is odd } E = { x ∈ Z: x is even } A) O and E form a partition of Z. B) O and E are not disjoint. C) O ∪ E ≠ Z

disjoint

Two sets, A and B in which their intersection is empty (A ∩ B = ∅)..

empty string

Unique string whose length is 0 and is usually denoted by the symbol λ.

E, A, C, D, and B.

Use the definitions below to match equivalent sets. A = { 1, 2, 4, 11, 12 } B = { 2, 4, 7, 8, 11 } C = { x ∈ Z: x is even } D = { x ∈ Z: x is odd } 1) B ∩ D A) { 2, 4, 11 } B) { 2, 4, 12 } C) Z D) { 1, 2, 4, 7, 8, 11, 12 } E) { 7, 11 } 2) A ∩ B A) { 2, 4, 11 } B) { 2, 4, 12 } C) Z D) { 1, 2, 4, 7, 8, 11, 12 } E) { 7, 11 } 3) C ∪ D A) { 2, 4, 11 } B) { 2, 4, 12 } C) Z D) { 1, 2, 4, 7, 8, 11, 12 } E) { 7, 11 } 4) A ∪ B A) { 2, 4, 11 } B) { 2, 4, 12 } C) Z D) { 1, 2, 4, 7, 8, 11, 12 } E) { 7, 11 } 5) A ∩ C A) { 2, 4, 11 } B) { 2, 4, 12 } C) Z D) { 1, 2, 4, 7, 8, 11, 12 } E) { 7, 11 }

B, E, A, C, D.

Use the definitions below to match equivalent sets. The universal set is Z. A = { 1, 2, 4, 11, 12 } B = { 2, 4, 7, 8, 11 } C = { x ∈ Z: x is even } D = { x ∈ Z: x is odd } D - B A) { 12 } B) The set of odd numbers except 7 and 11 C) Z D) { 1 } E) { 1, 11, 12 } A - ( B ∩ C ) A) { 12 } B) The set of odd numbers except 7 and 11 C) Z D) { 1 } E) { 1, 11, 12 } ( A - B ) ∩ C A) { 12 } B) The set of odd numbers except 7 and 11 C) Z D) { 1 } E) { 1, 11, 12 } C ⊕ D A) { 12 } B) The set of odd numbers except 7 and 11 C) Z D) { 1 } E) { 1, 11, 12 } *Line on top of B u C* ( B ∪ C ) ∩ A A) { 12 } B) The set of odd numbers except 7 and 11 C) Z D) { 1 } E) { 1, 11, 12 }

It indicates that the set is infinite.

What does (...) indicate in a set?

It indicates that an element is in set.

What does ∈ mean?

It indicates that an element is not in a set.

What does ∉ mean?

5

What is the length of string zzyzx?