You may be faced with questions that ask you to find the **union** or **intersection** of a set of numbers or lines. Usually, set notation is used to demonstrate the relationship between elements of information. A set is simply a group of things, such as numbers, objects, ideas, places, or geometric shapes. The items in the group are called elements and are surrounded by curly braces or { }.

The following are examples of sets:

- A = {pencil, notebook, paper}
- B = {1, 2, ,3, 4, 5}
- C = {a, b, c, d, e, f, g, h, i, j ,k}

The first three sections of this lesson (Elements, Unions/ Intersections, Subsets) will cover the material about sets that is likely to appear on the ISEE, while the last section will provide additional reference material for those who want to learn more.

The elements in a set are the items, values, or geometric values in a set.

You can represent that an element is part of a set by using the notation, **∈**.

Likewise, you can represent that an element is **not** part of a set by using the notation, **∉**.

The **union of sets** combines all elements from both sets into a single set, but values are not repeated.

Union is represented using the notation, **∪**.

In the union of *A* and *B*, pencil is not repeated; likewise, the union of *C* and *D* does not contain repeats of 3, 4, and 5, and the union of line segments *AC* and *BD* does not repeat segment *BC*. This is because typically sets are defined only by the number of unique elements contained, and thus do not contain duplicate elements. The set {11, 6, 6} is considered the same as the set {11, 6}.

The i**ntersection of sets**, on the other hand, comprises the elements shared by both sets, much like the intersection of two lines is a single point.

Intersection is represented using the notation, **⋂**.

The only shared element of sets *A* and *B* is pencil, while sets *C* and *D* share the elements 3, 4, and 5. Line segments *AC* and *BD* share segment *BC*.

Subsets are the sets that are contained within a larger set. The number of subsets a set has is determined by taking the number 2 to the power of the number of elements in the set.

*S* = 2^{n}

**Empty Sets**

- This formula includes the empty set { }, which is somtimes denoted as ∅.
- The empty set is the set containing no elements.
- There is only one empty set, which is a subset of any set.

Using this formula, we can evaluate the number of subsets in a set:

What does it mean to say that set A has 8 subsets? The subsets of set A are:

- {pencil}
- {notebook}
- {paper}
- {pencil, notebook}
- {pencil, paper}
- {notebook, paper}
- {pencil, notebook, paper}
- { }

Thus, we count 8 subsets of set A.

Though you will likely only be tested on elements, subsets, union, and intersection with set notation, there are many symbols involving sets that have specific actions and meanings.

Review this chart of symbols in set notation and their definitions:

**Answers to Practice Problems**

- Question 1
- Set A has 5 elements. Set B has 6 elements.
- Set A has 32 subsets. Set B has 64 subsets.
- {2, 4, 6, 7, 8, 9, 10, 11, 12}
- {8, 10}

- Question 2
- Set X has 7 elements. Set Y has 7 elements.
- Set X has 128 subsets. Set Y has 128 subsets.
- {a,b,c,d,e,f,g,i,k,m}
- {a,c,e,g}

- Question 3
- Set L has 6 elements. Set M has 7 elements.
- Set L has 64 subsets. Set M has 128 subsets.
- {binder, book, calculator, paper, lunch, quizbook, pen, pencil, sketchpad, timer}
- {book, calculator, quizbook}

- B
- B
- B

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