Set Theory- Math 121

Concerning Equality of Sets

1. If the two sets are equal, they contain exactly the same elements. 2. It then follows that each is a subset of the other. 3. If A=B, then every element of A is an element of B, and every element of B is an element of A. 4. In this case, A is an improper subset of B, and B is an improper subset of A.

Rules of Roster Notation

1. Listing each element of the set inside a pair of set braces. 2. In listing the members, each distinct member is listed only once. 3. In listing the members, the order in which the elements are listed matters not.

What is a subset?

1. When every element of one set, A, is also a member of a second set, B. 2. We say that the first set, A, is a subset of the second set, B. 3. In general, every set is a subset of itself. 4. In general, for any set set A, the empty set is a subset of the set.

Intersection of sets What is the set-builder's notation for set A and set B intersecting?

A intersection B Look it up

What is a set? The objects in the set Well-Defined Sets

A set is a collection of objects. The objects in the set are called elements or members. For example The set of all state capitals The set of all students in Math 121 The set of all even numbers that are positive

Union of sets What is the set-builder's notation for A union

A union B Look it up.

What is a universal set?

A universal set, denoted by U, is the set of all possible elements in that set.

What is an empty set?

An empty set is a set containing no elements.

Set-Builder Notation 1

It is sometimes considered inappropriate or impossible to describe a set in roster nation. For example The set M of all citizens of America, who are older than sixty, weigh less than sixty pounds, and live in the state of Illinois.

Set-Builder Notation 2

It lists the rules that determine whether an object is an element of the set rather than the actual elements. The "set of all x such that x is less than zero and x is a real number."

Does Not Belong To; Is Not An Element Of; Is Not In

Not Symbol

2.1

Sets and Set Operations

Belongs to; Is an element of; Is in.

Symbol

Cardinal number of a set, A

The cardinal number of a set, A, is the number of elements in the set, and it is denoted by n(A).

Determining Well-Defined Sets

The following are examples of not-well defined sets: 1. The set of all great tennis Hall-of-Fame Members 2. The set of all wrestling Hall-of-Fame members. The above sets are not well-defined because it is possible for someone to say that "The Rock" is not a WWE Hall of Famer, or that Novak Djokovic does not belong to the Tennis Hall of Fame> Hence, an opinion can not be a well-defined set.

Reading Set-Builder Notation

The set of all students such that the person is a student of Knox College and lives in the state of Louisiana.

Equality of Sets

Two sets are equal if and only if they contain exactly the same elements. The order in which the elements are listed does not matter.

When are two sets mutually exclusive?

Two sets are mutually exclusive (or disjoint) when they have no elements in common.

What is the difference between an improper and a proper subset?

You should be able to recognize this from the sign of the subset symbol.

The essential equation

n(A union B) = n(A) + n(B) - n(A intersection B)