The Algebra of Sets: Properties & Laws of Set Theory

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  • 0:04 Sets in Real Life
  • 0:25 Set Theory
  • 1:58 The Laws of Sets
  • 4:47 Lesson Summary
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Lesson Transcript
Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

The algebra of sets is an analysis of values. This lesson provides an overview of the properties of sets and laws of set theory and illustrates them with real-life examples.

Sets in Real Life

Do you have a favorite meal? Maybe it's a cheeseburger meal from your favorite hamburger restaurant. This meal probably includes a cheeseburger, French fries, a drink, some ketchup packets, and napkins. In real life, this is what we call a set.

The technical definition of a set is a collection of very specific objects. Let's go through the properties and laws of set theory in general.

Set Theory

A set of anything has to have specific criteria and be well defined. For example, one person may think that a cheeseburger dinner from a fast food restaurant is amazing, while someone else might be repulsed by its taste, making the criteria invalid. An example of a valid set would be edible foods that include bread, so the cheeseburger dinner would qualify. Let's make a list of foods and determine which ones are eligible for a set of edible items that include bread; we'll call our set ''S.''

S = {sandwich, hamburger, cheeseburger, toast, bread pudding}

The symbol ∈ indicates that something is part of a set. For example, grilled cheese ∈ S means that grilled cheese is part of set S. This is a true statement because grilled cheese is a sandwich. Ice cream ∉ S means that ice cream is not part of set S because it doesn't include bread.

Let's take a look at the properties of sets. The order of items in a set doesn't matter. In our set of edible foods that include bread, we could list toast first and sandwich last. If an item in a set is repeated, count it once. For example, let's say we have a set W that represents the letters in the word ''cheeseburger''. The example is here: W = {c, h, e, e, s, e, b, u, r, g, e, r}. As there are four e's and two r's, we can rewrite the set as W = {c, h, e, s, b, u, r, g}.

The Laws of Sets

Let's take a look at the different laws of sets one at a time.

1. Union of Sets

Let's say that we have two sets: S = {sandwich, hamburger, cheeseburger, toast, bread pudding} and B = {hamburger, cheeseburger}. We'll refer back to these sets throughout the rest of the lesson. The union of these sets is all items that are part of both sets, or ∪. The union of sets S and B is written as A ∪ B = {sandwich, hamburger, cheeseburger, toast, bread pudding}, which includes all of the items in both sets, but only one of each item if there are multiples.

2. Intersection of Sets

The intersection of sets defines what is common to both sets. For instance, in sets S and B, the hamburger and cheeseburger are common to both sets. The intersection of these sets is S ∩ B = {hamburger, cheeseburger}. This notation is similar to a Venn diagram of the two sets.

Hamburger and cheeseburger are included in both sets; therefore, they are the intersection of the two sets.

3. Commutative Law

Addition is a commutative property because 4 + 3 = 7 and 3 + 4 = 7; the order in which the numbers are added doesn't matter. This is true of the commutative law of sets too. For instance, S ∪ B is the same as B ∪ S. Likewise, S ∩ B is the same as B ∩ S.

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