SETS

A set is a collection of objects. You can have a set of books What is a set? A set is a collection of objects. You can have a set of books a set of china a set of pool balls …

Elements of a set The individual members of the set are called the elements of the set. In set notation, means that 2 is an element of the set of numbers {1,2,3} 4 is not a member of that set

How do you describe a set? If you have a set of numbers, there are two ways to describe the set Use a roster Use set-builder notation

Team A = {Andrews, Baxter, Jones, Smith, Wylie} The Roster Method Just like in gym class when they read the roster, this simply lists all of the object in the set. Team A = {Andrews, Baxter, Jones, Smith, Wylie}

When to use a roster A roster works fine if you only have a few elements in the set. However, many sets of numbers are infinite. Listing each member would take the rest of your life!

When to use a roster If an infinite set of numbers has a recognizable pattern, you can still use a roster. First establish the pattern then use an ellipsis … to indicate that the pattern goes on indefinitely. { 1, 2, 3, 4…}

Set-builder notation For many infinite sets it is easier to just use a rule to describe the elements of the set. We use brackets { } to let everyone know that we’re talking about a set. We use a vertical line | to mean “such that” let’s see how it works…

Set-builder notation { x | x > 0 } The set of all x such that x is positive. We couldn’t possible use a roster to describe this set because it includes fractions and decimals as well as the counting numbers. We couldn’t set up the pattern to begin with!

Subsets Set A is a subset of set B if all of the members in set A are also in set B. set A = { 1, 2, 3 } set B = { 1, 2, 3, 4, 5, 6 } Set A is a subset of set B

By definition, every set is a subset of itself. Subsets By definition, every set is a subset of itself. If set A has fewer elements than set B, it is called a proper subset and we use If they are the same set, it is called an improper subset and we use

Union Two people get married (union) and merge their DVD collections. They sell all the duplicates on eBay; the DVD’s that are left are the union of their collections. all of hers + all of his – dups Let’s look at a numeric example…

Union Set A = { 1, 2, 3 } set B = { 3, 4, 5 }

Intersection The intersection of Kirkwood and Walnut gets re-paved when Walnut get a facelift (because it’s part of Walnut) and it gets re-paved when Kirkwood get re-done (because it’s part of Kirkwood). The intersection of two sets are the elements that are members of both sets.

Intersection Set A = { 1, 2, 3 } set B = { 3, 4, 5 }

The empty set Set A = { 1, 2, 3 } set B = { 4, 5, 6 } There is no element that is in both sets, so the intersection is the empty set.

The empty set The symbol for the empty set is the Greek letter Phi Do not put it in brackets. is not empty. It is a set with one element – the Greek letter Phi

Vocabulary Element Subset Union Intersection Empty set