Discrete Math I Chapter 2 Sets and Functions

A⊕ B is equivalent to what expression?

(A-B) U (B-A)

What is a set

A collection of objects?

The _______of a finite set A, denoted by______ is the number of elements in A?

The cardinality of a finite set A, denoted by |A|, is the number of elements in A. If A = { 2, 4, 6, 10 }, then |A| = 4. The cardinality of the set B = { 1, 3, 5, ... , 99 } is 50. The cardinality of the empty set |∅| is zero.

The intersection of A and B?

The intersection of A and B, denoted A ∩ B and read "A intersect B", is the set of all elements that are elements of both A and B. The following animation illustrates the use of the intersection operation:

What is the power set?

the set of all subsets of a set denoted P(A)

How do you denote an ordered triple?

(x, y, z)

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 proper subset of B, denoted as A ⊂ B

If every element in A is also an element of B, then

A is a subset 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 subset of B, denoted as A ⊈ B.

A number x is _____________ if x ≥ 0.

A number x is non-negative if x ≥ 0.

A sequence of sets, A1, A2, ..., An, is pairwise disjoint if

A sequence of sets, A1, A2, ..., An, is pairwise disjoint if 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).

Let A,B,C be any finite sets. Give a concise description for which elements from A, B, C are in A⊕ B⊕ C.

An element x is in A⊕ B⊕ C if it is in exactly one or all 3 of the sets.

For n ≥ 4, an ordered list of n items is called an ________.

An ordered n-tuple. For example, (w, x, y, z) is an ordered 4-tuple and (u, w, x, y, z) is an ordered 5-tuple

What is the symmetric difference between two sets?

Denoted by A⊕B, is the set of elements that are a member of exactly one of A & B but not both!

What is Set Difference between two sets?

Difference between two sets, say A & B, denoted by A-B is the set of elements in A but not in B.

What are ellipses (...) used to indicate in set notation?

Ellipses are used to denote a long (possibly infinite) sequence of numbers. However, the sequence's pattern should be clear so that the reader can infer which elements are missing. B = { 1, 3, 5, ... , 99 } C = { 3, 6, 9, 12, .... }

A1, A2, ...,An is a partition for a non-empty set A if all of the following conditions hold:

For all i, Ai ⊆ A. For all i, Ai ≠ ∅ A1, A2, ...,An are pairwise disjoint. A = A1 ∪ A2 ∪ ... ∪ An

What is the set complement operation?

Given a universal set, A compliment is the set of all elements in U but not in A.

What is the special notation allows for a compact representation of the INTERSECTION of a long sequence of sets A1, A2, ..., An.

In the notation below, the expression "i = 1" below the intersection sign and the "n" above the intersection sign indicates that the intersection operation will be applied to all sets with integer indices i ranging from 1 through n.

What is the special notation allows for a compact representation of the UNION of a long sequence of sets A1, A2, ..., An.

In the notation below, the expression "i = 1" below the union sign and the "n" above the union sign indicates that the union operation will be applied to all sets with integer indices i ranging from 1 through n.

Cardinality of the power set

Let A be a finite set of cardinality n. Then the cardinality of the power set of A is 2n, or |P(A)|=2n.

What are the common mathematical sets?

N - The set of all natural numbers. All int >= 0 (0, 1, 2...) Z - The set of all integers (-2, -1, 0, 1, 2) Q - The set of all rational numbers:All real numbers that can be expressed as a/b(0, 1/2, 5.23, -5/3) where a and b are integers and b ≠ 0. & b ≠ 0. R - The set of real numbers. [0, 1/2, 5.23, -5/3, π, sqrt(2) ].

The union of two sets

The set of elements that are in either or both of the sets, denoted by U.

The set with ________ is called the empty set and is denoted by the symbol _______?

The set with no elements is called the empty set and is denoted by the symbol ∅

The _________ is used to indicate the positive elements of a particular set

The superscript + is used to indicate the positive elements of a particular set. For example, the set R+ is the set of all positive real numbers, and Z+ is the set of all positive integers.

The _______ is used to indicate the negative elements of a particular set

The superscript - is used to indicate the negative elements of a particular set. For example, the set R- is the set of all negative real numbers, and Z- is the set of all negative integers.

The symbol _____ is used to indicate that an element is in a set, as in 2 ___ A?

The symbol ∈ is used to indicate that an element is in a set, as in 2 ∈ A. The symbol ∉ indicates that an element is not in a set, as in 5 ∉ A.

Two sets are equal if

They have the same elements D = { 3, 4, 5 } E = { 5, 3, 4 } F = { 5, 3, 4, 6 } D = E

A = B if and only if

Two sets are equal if and only if each is a subset of the other: A ⊆ B and B ⊆ A

Two sets, A and B, are said to be disjoint if

Two sets, A and B, are said to be disjoint if their intersection is empty (A ∩ B = ∅).

What is the universal set?

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

What is the common way of finding whether or not a particular set is a subset of another set?

Venn Diagrams

A partition of 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.

What are elements?

objects in a set