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Set Theory: Definition & Application

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Set theory is a branch of mathematics that is concerned with groups of objects and numbers known as sets. Set theory deals with the properties of these sets as single units, regardless of the nature of the individual elements within the set.

Set Theory Definitions

Emma and her sister Michelle both want to open pie shops. Emma wants to serve five kinds of pie: chocolate, key lime, strawberry, cherry, and peach. Michelle wants to serve five kinds of pie as well, but not exactly the same ones that Emma plans to sell. She is going to sell strawberry, peach, apple, pear, and lemon pies.

Each of these groups of pies can be considered to be a set. In set theory, a set is collection of objects, and each item in the set is called an element.

In this case, there are two sets. Let's call them E and M (for Emma and Michelle). Each set contains exactly five elements. Even though most sets do contain at least one element, there are special cases when a set contains NO elements. A set containing exactly zero elements is called an empty set.

We can further define certain elements within a set to be in a subset, which is a smaller set of elements taken from within a set. For example, the subset of Set E that only contains fruit pies would contain four elements: key lime, strawberry, cherry, and peach. Chocolate would not be in the subset even though it is an element of the larger set.

pie sets

If we combine these two sets, this is called a union, and the symbol of a set union looks like the letter U. In this case, the union of sets E and M contains eight elements: chocolate, key lime, strawberry, cherry, peach, apple, pear, and lemon.

pie set union

The intersection of the two sets contains only those elements that are found in BOTH sets. Therefore the intersection of sets E and M contains only two elements: strawberry and peach. The symbol for an intersection looks like an upside down U.

pie set intersection

The complement to a set contains all the elements that are NOT in the set. For example, the complement to set M would contain every kind of pie except for those already in the set. Finally, disjointed sets are sets that do not have any elements in common. The intersection of two disjointed sets would be an empty set.

Sets Containing Numerical Data

The elements of a set can numerical or non-numerical, like the pies in the previous example. Can you think of some examples of set containing numerical data?

Let's say that set A contains all odd numbers less than or equal to 25. This means that set A would contain the elements 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, and 25. We could also define another set, set B, that contains all prime numbers less than or equal to 25. Set B would contain the elements 2, 3, 5, 7, 11, 13, 17, 19, and 23.

Which elements would be in a set formed from the intersection of set A and set B? Remember that the intersection of two sets contains elements that are common to both sets. In this case, the intersection would contain the elements 3, 5, 7, 11, 13, 17, 19, and 23. The elements 2, 9, 15, and 25 would NOT be in this new set because they are not found in both set A AND set B.

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