Learning Objectives
- Universal Set
- Using sets and proper notation, perform the operations of
- complement
- union
- intersection
- Order of Set Operations
- Cardinality of Set Operations
- Equal and Equivalent Sets
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.
Math vocabulary and notation
It takes repetition and practice to obtain new vocabulary in a language that is new to you. Math is no different in many respects than learning a new language, with all its vocabulary, syntax, and spelling conventions. The symbols in this section may be completely unfamiliar to you. If so, you’ll need to spend time with them, employing flashcards and writing them out by hand in context.
Give yourself time to learn and appreciate the language of mathematics!
Universal Set
A universal set is a set that contains all the elements we are interested in. This would have to be defined by the context.
Example
- If we were discussing searching for books, the universal set might be all the books in the library.
- If we were grouping your Facebook friends, the universal set would be all your Facebook friends.
- If you were working with sets of numbers, the universal set might be all whole numbers, all integers, or all real numbers
Complement
The complement of a set A contains everything that is not in the set A. The complement is notated A’.
A complement is relative to the universal set, so A’ contains all the elements in the universal set that are not in A.
a new use for a superscript
Notice in the descriptions of the notation introduced above that the complement of a set is denoted A′. This superscript is not an exponent, it is a superscript (superscripts can be for variables). It is a symbol that denotes the complement of a set.
Example
Suppose the universal set is U = all whole numbers from 1 to 9. If A = {1, 2, 4}, then A’ = {3, 5, 6, 7, 8, 9}.
Try It
Union and Intersection
The union of two sets contains all the elements contained in either set (or both sets). The union is notated A ⋃ B. More formally, x ∊ A ⋃ B if x ∈ A or x ∈ B (or both)
The intersection of two sets contains only the elements that are in both sets. The intersection is notated A ⋂ B. More formally, x ∈ A ⋂ B if x ∈ A and x ∈ B.
intersection and union symbols
The intersection [latex]\cap[/latex] and union [latex]\cup[/latex] symbols look a little like letters in the alphabet. In fact, that’s a trick for remembering them.
The union symbol looks like a capital U, for union.
The intersection symbol looks a little like a big lower-case n, for in-tersect
Example
Consider the sets:
A = {red, green, blue}
B = {red, yellow, orange}
C = {red, orange, yellow, green, blue, purple}
Find the following:
- Find A ⋃ B
- Find A ⋂ B
- Find A’⋂ C
Notice that in the example above, it would be hard to just ask for A’, since everything from the color fuchsia to puppies and peanut butter are included in the complement of the set. For this reason, complements are usually only used with intersections, or when we have a universal set in place.
Try It
Order of Set Operations
As we saw earlier with the expression A’ ⋂ C, set operations can be grouped together. Grouping symbols can be used like they are with arithmetic – to force an order of operations.
Example
Suppose H = {cat, dog, rabbit, mouse}, F = {dog, cow, duck, pig, rabbit}, and W = {duck, rabbit, deer, frog, mouse}
- Find (H ⋂ F) ⋃ W
- Find H ⋂ (F ⋃ W)
- Find (H ⋂ F)’ ⋂ W
How does Cardinality apply to Set Operations
Exercises
Let A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}.
What is the cardinality of B? A ⋃ B, A ⋂ B?
Try It
Equal and Equivalent Sets
Two sets are equal when they have the same elements. They might not be in the same order.
For example: {pink, purple, blue, green} is equal to {green, blue, purple, pink}
Two sets are equivalent when they have the same number of elements, not necessarily the same elements.
For example: {June, July, August} is equivalent to {4,8,9} because they each have 3 elements.
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