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Calculating Unions & Intersections in Mathematical Sets

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will review the set operations of unions, intersections, and differences on sets. We will explore these operations with Venn diagrams and look at some properties these operations satisfy.

Mathematical Sets

It is rare to find sisters that don't share clothes! Suppose that Lacy and Anne are sisters. They wear the same size in everything, so they will often split the cost of a certain item of clothing and share it as though they both owned it. Because of this, all of their clothes fall into different categories.

  1. Clothes they shared the cost for, so both own.
  2. Clothes that one of the girls bought themselves, so only they own it.
  3. Clothes that were bought by either or both of the girls, so are owned by either or both of the girls.

Contrary to what you might think, these different categories actually have mathematical significance! If we think of all of the clothes in the world as a set, call it U; the clothes that Lacy owns as one set, call it L; and the clothes that Anne owns as another set, call it A; then we have the following facts:

  • Category 1 is the set of all clothes that are both in set L and in set A.
  • Category 2 is the set of all clothes in set L but not in set A, or in set A but not in set L.
  • Category 3 is the set of all clothes in either of the sets L or A.

In mathematics, set operations are operations that we use to create new sets from given sets. In this lesson, we will look at three operations: unions, intersections, and differences.

Operations of Mathematical Sets

If we have a set of objects, called the universal set, and two sets of objects, call them sets R and S, within the universal set, then the set consisting of the objects that are in both sets R and S is called the intersection of sets and is denoted as RS. The set consisting of the objects in one set, say R, but not the other, S, is called the difference of sets and is denoted as R - S. The set consisting of all the elements in sets R and S is called the union of sets and is denoted as RS.

In our example, all of the clothes in the world would be the universal set. Category 1 consists of clothing in both sets L and A, so would be LA. Category 2 consists of clothing in one set but not the other, so would be L - A or A - L. Category 3 consists of clothing in either L or A, so would be LA.

Using Venn Diagrams to Perform Operations of Sets

When trying to perform operations on sets, we can use a tool called Venn diagrams. These diagrams give us a visual representation of sets and allow us to more easily calculate unions, intersections, and differences. To illustrate a Venn diagram, let's use an example.

Suppose we have a universal set of the following numbers:

  • U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Now, within that set, consider the following two sets:

  • T = multiples of 2 in the universal set = {2, 4, 6, 8, 10}
  • F = multiples of 4 in the universal set = {4, 8}

In a Venn diagram, we would represent the universal set with a large rectangle and we would represent the sets within the universal set as circles.


unint1


The intersection of two sets would be where their circles intersect. The union of two sets would be everything in either of their circles.


unint2


Using this diagram, we can place each of the elements in the universal set in the appropriate area. For example, consider the numbers 1, 2, and 4.

  • The number 1 is in the universal set but is not a multiple of 2 or 4, so it would go in the rectangle but outside of the circles representing sets T and F.
  • The number 2 is in the universal set and is a multiple of 2 but is not a multiple of 4, so it would go in the circle T but not in the circle F.
  • The number 4 is in the universal set and is a multiple of both 2 and 4, so it would go in both circles T and F, or where the circles intersect.

We can place all the numbers in the universal set in the Venn diagram using this logic.


redo1


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