3.8 Subsets ( )

Some or ALL Elements of a given set

Ex: U { Natural numbers }

We say that A is a subset of U and we write it as: A U

→U = { 1,2,3,….} →A = { 1,2,3,4,5,6,7,8,9}A { x | 0 < x ≤ 9 }

Union (U) : The UNION of two or more sets is the set that contains ALL the elements of the setsEx: U { Natural numbers } A { x | 0 < x ≤ 9 } B { x | 0 < 2x < 12}

1, 2, 3, 4, 5, 6, 7, 8, 9

2, 4, 6, 8,10

NOTICE: we must never repeat elements in a union of sets, So what do we DO?

A U B =VENN DIAGRAM

→U = { 1,2,3,…∞} →A = { 1,2,3,4,5,6,7,8,9}

→B = { 2,4,6,8,10}

Union (U)

1, 2, 3, 4, 5, 6, 7, 8, 9

2, 4, 6, 8, 10

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

INTERSECTION (∩)

The INSTERSECTION of two or more sets is the set of elements that are COMMON to every set. ( elements that belong to ALL the sets)

Ex: U { Natural numbers } A { x | 0 < x < 9 } B { x | 0< 2x < 9}

INTERSECTION (∩)

0, 1, 3, 5, 7, 9

2, 4, 6, 8

Thus A ∩ B = { 2, 4, 6, 8 }

Ex: U { Whole numbers } A { x | 0 ≤ x ≤ 9 } B { x | 0 < 2x < 13}

10, 12

→A = { 0,1,2,3,4,5,6,7,8,9}→B = { 2,4,6,8,10,12}

COMPLEMENT: (Elements not in A ∩ B) (A ∩ B ) ‘ = { 0, 1, 3, 5, 7, 9, 10, 12}

CROSS PRODUCT (X)

The CROSS PRODUCT of two or more sets is found by using the distributive property. You pair the elements of the firsts with the elements of the second.

Ex: A { a, b } B { 1,2,3 }

CROSS PRODUCT (X)

A X B = {a, 1} {b,1} {a, 2} {b,2} {a, 3} {b,3}

DISJOINT : Sets that have nothing in common.

0, 1, 2, 3, 4, 5, 6, 7, 8,9

Thus A and B are Disjointed

Ex: U { Whole numbers } A { x | 0 ≤ x ≤ 9 } B { x | 10 < 2x < 18}

12, 14,16

→A = { 0,1,2,3,4,5,6,7,8,9} →B = { 12,14,16}

GOAL:

REAL-WORLD:Three friends are going camping. The items in each backpack form a set. What is the intersection of items of the backpacks? Create a Venn Diagram.

flashlightmappansunglasseswater

camerafirst aid kithat mapwater

First aid kitmaphatpanropewater

SOLUTION: First look at what they have in common (intersection):

mapwater

hat

flashlight

sunglasses

camera

ropepan

first aid kit

YOU TRY IT:A = { x | x is one of the five letters in the English alphabet}B = { x | x is a vowelC = { x | x is a letter in the world VEGETABLE}

Provide a Venn-Diagram to show the intersection of the three sets.

SOLUTION:A = { a, b, c, d, e}

Furthermore:

B = { a, e}

C = { V, E, G, E, T, A, B, L, E}

AB= { a,e}

AC= { a,e,b}

BC= { a,e}

SOLUTION: First look at what they have in common (intersection):

eva

b

ct

g

d l

A C

B

VIDEOS: Sets

Sets:

http://www.khanacademy.org/math/probability/independent-dependent-probability/basic_set_operations/v/intersection-and-union-of-sets

CLASSWORK:

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Problems: 1, 3, 5, 6, 7, 10, 2425, 35, 36, 45.