• Understand set notation and basic set theory concepts

Applying Set and Interval Notation

Set notation and interval notation are crucial tools in mathematics for describing ranges of values precisely and concisely. These notations assist in expressing solutions to equations, defining function domains and ranges, and setting up integration and summation intervals.

Set Notation generally refers to a way of describing sets with a list of elements or a rule that the elements must follow, often expressed in the form [latex]\{x | \text{ property of } x\}[/latex]. This can include descriptions of sets that are not continuous, such as [latex]\{1, 2, 3, 4\}[/latex] or [latex]\{x | x \text{ is an even integer}\}[/latex].

Interval Notation focuses on describing continuous intervals on the real number line, often using parentheses and brackets, such as [latex](a, b)[/latex] or [latex][c, d)[/latex]. In interval notation:

• We use parentheses ( ) for intervals to indicate that the endpoint is not included, known as an open interval.
• We use brackets [ ] for intervals to indicate that the endpoint is included, known as a closed interval.
• The union symbol [latex]\cup[/latex] is used to combine disjoint sets or intervals that are part of the domain or range but do not directly connect.
• Infinity (∞) is always accompanied by a parenthesis because infinity is not a number but rather a concept of endlessness.

set and interval notation

• Set Notation: Uses curly braces [latex]\{\}[/latex] to list elements explicitly or to describe them with conditions.
• Interval Notation: Efficient for describing continuous ranges, using brackets [latex][][/latex]for closed intervals and parentheses [latex]( )[/latex] for open intervals.

Some examples of set and interval notation being used are:

1. Describing a Domain: The domain of [latex]f(x)=\sqrt{x−3}[/latex] is all [latex]x[/latex] such that [latex]x−3≥0[/latex]. In interval notation, this is [latex][3,∞)[/latex].
2. Solution Sets: For the inequality [latex]x^2−4<12[/latex], solve to find [latex]x<4[/latex] and [latex]x>−4[/latex], described as [latex](−4,4)[/latex].
3. Defining Function Ranges: If a function [latex]f[/latex] maps real numbers to their squares, the range can be set as [latex][0,∞)[/latex], representing all non-negative real numbers.

Always determine whether the interval should include the endpoints based on the conditions given.

Write the following sets using interval notation.

1. The set of all real numbers greater than [latex]5[/latex]
2. [latex]{x∣x≤−2}[/latex]
3. [latex]{x∣x≤−3 \text{ or } x≥3}[/latex]

Show Answer

1. This set includes every real number greater than [latex]5[/latex], but not [latex]5[/latex] itself. In interval notation, you represent this as [latex](5,∞)[/latex].
2. The set describes all real numbers less than or equal to [latex]-2[/latex]. In interval notation, this is written as [latex](−∞,−2][/latex].
3. This set includes all real numbers less than or equal to [latex]-3[/latex] and all real numbers greater than or equal to [latex]3[/latex]. In interval notation, these two conditions are represented as two separate intervals combined using the union symbol. Thus, the notation for this set is [latex](−∞,−3]∪[3,∞)[/latex].

Set Operations

Commonly, sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.

set operations: union, intersection, complement, and difference

• The union of two sets contains all the elements contained in either set (or both sets).
  • The union is notated [latex]A \cup B[/latex]. More formally, [latex]x \in A \cup B[/latex] if [latex]x \in A[/latex] or [latex]x \in B[/latex] (or both).
• The intersection of two sets contains only the elements that are in both sets.
  • The intersection is notated [latex]A \cap B[/latex]. More formally, [latex]x \in A \cap B[/latex] if [latex]x \in A[/latex] and [latex]x \in B[/latex].
• The complement of a set [latex]A[/latex] contains everything that is not in the [latex]A[/latex].
  • The complement is notated [latex]A'[/latex], or [latex]A^c[/latex], or sometimes  ~[latex]A[/latex].
• The difference of two sets is the list of all the elements that are in one set but not present in the other.
  • The difference between two sets is notated [latex]A \setminus B[/latex]. More formally, [latex]x \in A \setminus B[/latex] if [latex]x \in A[/latex] & [latex]x \notin B[/latex].

Notice in the descriptions of the notation introduced above that the complement of a set is denoted [latex]A^{c}[/latex]. This superscript is not an exponent. It is a decoration that denotes the complement of a set.

Let’s try applying the set operations.

Consider the sets:

• [latex]A = \{\text{red, green, blue}\}[/latex]
• [latex]B = \{\text{red, yellow, orange}\}[/latex]
• [latex]C = \{\text{red, orange, yellow, green, blue, purple}\}[/latex]

Find the following:

1. Find [latex]A \cup B[/latex]
2. Find [latex]A \cap B[/latex]
3. Find [latex]A^c \cap C[/latex]

Show Solution

1. The union contains all the elements in either set: [latex]A \cup B = \{\text{red, green, blue, yellow, orange}\}[/latex]. Notice we only list red once.
2. The intersection contains all the elements in both sets: [latex]A \cap B = \{\text{red}\}[/latex].
3. Here we’re looking for all the elements that are not in set [latex]A[/latex] and are also in [latex]C[/latex] . [latex]A^c \cap C = \{\text{orange, yellow, purple}\}[/latex]

Set operations can be grouped together – for example, [latex]A^c \cap C[/latex]. Grouping symbols can be used like they are with arithmetic – to force an order of operations.