The Main Idea 

A set is a collection of unique objects or elements. These elements can be anything: numbers, people, other sets, etc. For example, the set [latex]A[/latex] could be the set of all positive integers, while the set B could be the set of all colors in the rainbow.

The empty set, also known as the null set, is a special set that contains no elements. For example, the set of all dogs with six legs is an empty set, since no such dogs exist.

Notation
In set theory, we often use curly brackets – [latex]\{\}[/latex] – to denote a set. An example of a set is [latex]A = \{a, b, c, d\}[/latex].

Commonly, we will use a variable to represent a set, to make it easier to refer to that set later. We usually denote sets with capital letters such as [latex]A[/latex], [latex]B[/latex], [latex]C[/latex], etc.

The symbol [latex]\in[/latex] means “is an element of”.

An empty set is denoted by the symbols is notated by [latex]\emptyset[/latex] or [latex]\{ \}[/latex].

The Main Idea 

A set [latex]A[/latex] is said to be a subset of another set [latex]B[/latex] if every element in [latex]A[/latex] is also in [latex]B[/latex]. We denote this by [latex]A \subseteq B[/latex]. For instance, if [latex]B = \{\text{red}, \text{green}, \text{blue}, \text{yellow}\}[/latex] and [latex]A = \{\text{red}, \text{blue}, \text{yellow}\}[/latex], then [latex]A[/latex] is a subset of [latex]B[/latex] because every color in [latex]A[/latex] is also in [latex]B[/latex].

A proper subset is similar to a subset, but with one key difference: if [latex]A[/latex] is a proper subset of [latex]B[/latex], then there is at least one element in [latex]B[/latex] that is not in [latex]A[/latex]. We denote this by [latex]A \subset B[/latex]. If [latex]B = \{\text{red}, \text{green}, \text{blue}, \text{yellow}\}[/latex] and [latex]A = \{\text{red}, \text{blue}\}[/latex], then [latex]A[/latex] is a proper subset of [latex]B[/latex].

To calculate the number of subsets and proper subsets in a set you must use the following formulas:

If you have [latex]n[/latex] elements in your set:

• Number of subsets: [latex]2^{n}[/latex]
• Number of proper subsets: [latex]2^{n}−1[/latex]

The Main Idea 

The union of two sets contains all the elements contained in either set (or both sets) and is notated [latex]A \cup B[/latex]. The formal notation states, [latex]x \in A \cup B[/latex] if [latex]x \in A[/latex] or [latex]x \in B[/latex] (or both).

If we break this down and translate the symbols into words it is easier to understand.

• [latex]x \in A \cup B[/latex] reads as “[latex]x[/latex] is an element of the union of sets [latex]A[/latex] and [latex]B[/latex] “
• [latex]x \in A[/latex] reads as “[latex]x[/latex] is an element of set [latex]A[/latex] “
• [latex]x \in B[/latex] reads as “[latex]x[/latex] is an element of set [latex]B[/latex] “

If we add all the parts we translated into words back into the original formal definition it reads as “[latex]x[/latex] is an element of the union of sets [latex]A[/latex] and [latex]B[/latex] if [latex]x[/latex] is an element of set [latex]A[/latex] or [latex]x[/latex] is an element of set [latex]B[/latex] (or both).”

The Main Idea 

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].

If we break this down and translate the symbols into words it is easier to understand.

• [latex]x \in A \cap B[/latex] reads as “[latex]x[/latex] is an element of the intersection of sets [latex]A[/latex] and [latex]B[/latex] “
• [latex]x \in A[/latex] reads as “[latex]x[/latex] is an element of set [latex]A[/latex] “
• [latex]x \in B[/latex] reads as “[latex]x[/latex] is an element of set [latex]B[/latex] “

If we add all the parts we translated into words back into the original formal definition it reads as “[latex]x[/latex] is an element of the intersection of sets [latex]A[/latex] and [latex]B[/latex] if [latex]x[/latex] is an element of set [latex]A[/latex] and [latex]x[/latex] is an element of set [latex]B[/latex]. “

The Main Idea 

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].

If we break this down and translate the symbols into words it is easier to understand.

• [latex]x \in A \setminus B[/latex] reads as “[latex]x[/latex] is an element of the difference of sets [latex]A[/latex] and [latex]B[/latex] “
• [latex]x \in A[/latex] reads as “[latex]x[/latex] is an element of set [latex]A[/latex] “
• [latex]x \notin B[/latex] reads as “[latex]x[/latex] is not an element of set [latex]B[/latex] “

If we add all the parts we translated into words back into the original formal definition it reads as “[latex]x[/latex] is an element of the difference of sets [latex]A[/latex] and [latex]B[/latex] if [latex]x[/latex] is an element of set [latex]A[/latex] and [latex]x[/latex] is not an element of set [latex]B[/latex] .“

The Main Idea 

A universal set is a set that contains all the elements we are interested in. This would have to be defined by the context.

The Main Idea 

A complement is relative to the universal set. The complement of a set [latex]A[/latex] contains everything that is not in the universal set [latex]A[/latex].  The complement is notated [latex]A'[/latex], or [latex]A^c[/latex], or sometimes  ~[latex]A[/latex].

The Main Idea 

Venn diagrams are a helpful tool for visualizing the relationships between sets and performing set operations. A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items. Each circle represents a set, with the items of the set represented by points within the circle. Here’s how you can create and interpret Venn diagrams for various set relationships and operations:

• Representing Sets: In a Venn diagram, each set is represented by a circle. For example, if you have two sets A and B, you can represent them with two circles. The universal set is often represented by a rectangle encompassing all the circles.
• Union: The union of two sets A and B (A ∪ B) is represented by the area covered by both circles A and B, including the overlap.
• Intersection: The intersection of two sets A and B (A ∩ B) is represented by the area where the two circles overlap.
• Difference: The difference of two sets A and B (A – B) is represented by the area that is in circle A but not in circle B.
• Complement: The complement of a set A (A’) is represented by the area that is outside circle A but within the universal set.
• Empty Set: An empty set or null set, which contains no elements, can be represented by a circle with nothing in it. If the set is a subset of another set, it can be represented by a circle within the larger set, but with no elements.