SETS

Elements of a set The individual members of the set are called the elements of the set. is an element of a set of dishes.

Elements of a set The symbol means “is an element of” (hint: looks like an e for element!) means that 2 is an element of the set of numbers {1, 2, 3}

means 4 is not an element of the set {1, 2, 3} Elements of a set Remember Ghostbusters? means 4 is not an element of the set {1, 2, 3}

A well-defined set A set is “well-defined” if it has clear rules that make it obvious if something is an element of the set or not.

A well-defined set? The set of the members of the 2007 Colts team is a “well-defined” set.

George Bush – a Colt? Even though he’s in the picture, you know George Bush is not really a member of the team because the 2007 Colts team is a “well-defined” set.

NOT a well-defined set The set of the 5 best Colts players in history is not a well-defined set because different people may have different opinions.

Cardinality of a set The cardinality (or cardinal number) of a set is just the number of elements in the set. The cardinality of {2, 4, 6} is 3.

Cardinality of a set The symbol for the cardinality (or cardinal number) of a set is n( ). If A = {2, 4, 6} then n(A) = 3.

Equivalent sets Two sets are equivalent if they have the same number of elements or the same cardinality n( ). If A = {2, 4, 6} and B = {1, 3, 5} n(A) = 3 and n(B) = 3. A and B are equivalent sets.

Equal sets Two sets are equal if they have exactly the same elements. Order doesn’t matter. If A = {2, 4, 6} and B = {6, 4, 2} A and B are equal sets.

Equal Sets are Equivalent If A = {2, 4, 6} and B = {6, 4, 2} A and B are equal sets. n(A) = 3 and n(B) = 3 A and B are equivalent sets

Equivalent Sets may NOT be Equal If A = {2, 4, 6} and B = {1, 3, 5} n(A) = 3 and n(B) = 3 A and B are equivalent sets but A and B are NOT equal sets.

Naming a set Sets are traditionally named with capital letters. Let N = {a, b, c, d, e}

Describing a set Set are typically enclosed in braces { } You can describe a set by using: words a roster set-builder notation

Describing a set The easiest way to describe many sets is by using words. B = {all blue-eyed blonds in this class} There isn’t a really good mathematical equivalent for that!

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…}

The Natural Numbers The set of natural numbers N is also called the set of counting numbers. N = { 1, 2, 3, …}

Finite Sets We think of a finite set as having a “countable” number of elements. Mathematically, that means that the cardinality of the set (number of elements) is a natural number {1, 2, 3, …} Geek Patrol: If A is a finite set, n(A) N

Set-builder notation For many infinite sets it is easier to just use a rule to describe the elements of the set. We use braces { } 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!

The Empty Set If a set has NO elements, we call it the empty set. The symbol that we will use for the empty set is The cardinal number for the empty set is 0.

OOPS! Notice that the empty set is the one set that we do not enclose in braces. is not empty! It is a set with one element – the Greek letter Phi.

Vocabulary Element Well-defined Cardinality Equivalent vs. equal Finite vs. infinite Roster vs. set builder notation Empty set