Set Operations & Cartesian Products

Unions

Notation;

Union of set A and B

{x

A Union of Sets is all the elements in both of the sets

Any element (x) is either an element of Set A and/or Set B {x| x

Find the Union { p, q, r, s, t, u} { r, s, t, u, v, w, x,

y} { p, q, r, s, t, u, v, w, y}

FIND• { 5 , 7 , 9 ,} {3 , 5 , 7 , 9 , 11 , 13}

• {3, 5, 7, 9, 11, 13}

• {22 , 33 , 44 , 55 , 66} {12 , 23 , 34 , 45 , 56}• {12, 22, 23, 33, 34, 44, 45, 55, 56, 66}

Examples; Unions of Sets

Intersections

Notation;

Intersection of Set A and Set B

{x

An Intersection of a set is the elements that both sets share

Any element (x) is both an element of Set A and Set B {x| x

Find the Intersection { p, q, r, s, t, u} { r, s, t, u, v, w, x,

y} { r, s, t, u}

FIND• { 5 , 7 , 9 , 11 , 13 ,15} {3 , 5 , 7 , 9 , 11 , 13 ,17 , 19}

• {5, 7, 9, 11, 13}

• {22 , 33 , 44 , 55 , 66} {12 , 23 , 34 , 45 , 56}

Examples; Intersection of Sets

Difference of Sets

Notation;

Difference of set A and B

{x

Just like regular subtraction. Start with the first set and take away elements that are in both sets

Any element (x) is any element of Set A that is Not an element of Set B {x| x

Find the Difference { p, q, r, s, t, u} { r, s, t, u, v, w, x,

y} { p, q}

FIND• {COW, P IG, GOAT, DONKEY, HORSE} – { GOAT, DONKEY,

COW}• {Pig, Horse}

• {22 , 33 , 44 , 55 , 66} {12 , 23 , 34 , 45 , 56}• { 22, 33, 44, 55, 66}

Examples; Difference of Sets

Order Matters! Different than sets

Use parentheses, not brackets

Ordered Pair (a,b) a= first component b= second component

(a,b) (b,a)

In sets, it does not matter what order the elements are in Ordered pairs are

equal only if both first components are the same and both second components are the same

Ordered Pairs

Cartesian Products

Named after Rene Descartes

• Formulated analytic Geometry

• French Philosopher

• “I think, therefore I am”

Cartesian Products of Sets

o Sets contain elements, and you can use those elements to form ordered pairso You can take elements from Set A and pair them with Set Bo A = {1, 2, 3} B = { 5, 10}

o Cartesian Product Notation;

oA x B = {(a,b) | a A and b B}o “A cross B” such that the ordered pair (a,b)

where a is an element of Set A and b is an element of Set B

End of Day 1

“ORDER MATTERS”

Cartesian Products

FindingCartesian productsNotation;

A x B = { (x,y) │ x A and y B}

The order in which the sets are listed matters. First component is always from

set A Second component is always from

Set B

Example N= {4, 3, 2} T = {1, 5} Find the Cartesian Product N x T Pair each element of N with each

element of T {(4,1), (4,5), (3,1), (3,5,), (2,1,),

(2,5)}

• Let Z = {5, 9, 3, 6}• Find Z x Z

• {(5,5), (5,9), (5,3), (5,6), (9,5), (9,9), (9,3), (9,6), (3,5),(3,9), (3,3), (3,6), (6,5), (6,9), (6,3), (6,6)}

Example: Cartesian Product of a set with itself

Cardinal Numbers of Cartesian Products

n(A) • n(B)

Let n(A) = a

Let n(B) = b

Multiply together the cardinal numbers for each set, to find the Cardinal Number (total # of elements) in the Cartesian Set

Order of sets doesn’t change cardinal number of the Cartesian Products n(A) • n(B) = n(B) • n(A)

• P = {2 ,5 ,7} Q = {8 , 9 , 10}• FIND P X Q

• {(2,8), (2,9), (2,10), (5,8), (5,9), (5,10), (7,8), (7,9), (7,10)}• n(P x Q) = 9 • n(P) =3 and n(Q) = 3• n(P) • n(Q) = 9

• FIND Q X P• {(8,2), (8,5), (8,7), (9,2), (9,5), (9,7), (10,2), (10,5), (10,7)}• n(Q x P) = 9

Cardinal Number of Cartesian Products

REPRESENT SET OPERATIONS WITH VENN DIAGRAMS

• COMPLIMENT• INTERSECTION

• UNION• DIFFERENCE

• CARTESIAN PRODUCT

Venn Diagrams

De Morgan’s Law

• And

• The compliment of the intersection of the two sets is equal to the union of the compliment of two sets

U= {a, b, c, d, e, f, g}A= { a, b, c) B= {c, d, e}

• {a, b, c, d, e}• ()’ = {f, g}

• { c}• )’ = {a, b, d, e, f, g}

• A’ = { d, e, f, g}• B’ ={ a, b, f, g}

SameSame

U

BA

a b cd e

f g