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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