Math 1320 Chapter 6: Sets and Counting 6.1 Sets and Set Operations

Getting Started Definitions – A set is a collection of items. These items are referred to as elements of the set. Fact: We usually use capital letters to represent the sets and lower case letter to indicate the elements. Therefore, a is an element of the set A.

Notation

Obvious(?) Facts A finite set has finitely many elements. – A finite set might be the set of students in this classroom. An infinite set does not have finitely many elements. – The set of natural numbers never ends so you cannot count them all and is therefore infinite.

Examples Some standard sets and what they contain: – A standard coin has a set of outcomes when flipped and noting the side facing up that is equal to {H, T}. – If two coins are tossed there are two possibilities: Indistinguishable = {HH, HT, TT} Distinguishable = {HH, HT, TH, TT} – For more sets involving dice, coins, and playing cards, see the handout titled “dice coin card” on my website.

More Notation

Definitions

Examples: List the elements of each set. 1.The set F consisting of the four seasons. 2.The set of outcomes of tossing two distinguishable coins. 3.The set of all outcomes of tossing two indistinguishable dice such that the sum of the numbers is 8. 4.The set of all outcomes of tossing two distinguishable dice such that the sum of the numbers is 8.

Examples: List the elements of each set. 1.The set F consisting of the four seasons. Solutions: F = {spring, summer, fall, winter} Note – the order (or language) doesn’t matter

Examples: List the elements of each set. 2. The set of outcomes of tossing two distinguishable coins. Answer: If the coins are distinguishable, lets make one a quarter and the other a penny. The set of outcomes is then {HH, HT, TH, TT}.

Examples: List the elements of each set. 3. The set of all outcomes of tossing two indistinguishable dice such that the sum of the numbers is 8. Answer: They are indistinguishable so we won’t know if we rolled a 6 and then a 2 or a 2 and then a 6. We will write each sum only one time to get {(2,6), (3, 5), (4,4)}.

Examples: List the elements of each set. 4. The set of all outcomes of tossing two distinguishable dice such that the sum of the numbers is 8. Answer: Now the dice are distinguishable. Let’s suppose one is blue and the other is red. Now the order they are rolled matters. We get the solution {(2,6), (3,5), (4,4), (5,3), (6,2)}

Venn diagram example Draw a Venn diagram that illustrates the relationships among the given sets. S = {eBay, Google, Amazon, OHagan, Hotmail} A = {Amazon, Ohagan} B = {eBay, Amazon} C = {Amazon, Hotmail}

Examples

Cartesian Products