Unit 1 Sets

Definitions (Day 1)

Sets: A collection of numbers or objects.

Element: Every number or object in set is called an element.

Notations

∈=means is anelement ofa set∉ = means not an element of a set∅=empty set .Theset with nothing .

Practice:

Write each word phrase in set notation.

(1) 10 is an element of set D

(2) 16 is not an element of set A

(3) 8 is not an element of the set of the first 7 prime numbers.

Write each of the following set notations in a complete sentence.

(4) 9 ∉ B (5) 21 ∈ C (6) 3.5 ∉ {0 ,±1 ,±2 ,±3 ,… }

Definition

Subset: If P and Q are two sets, then P is a subset of Q if every element of P is also an element of Q. Denoted as;

P ⊆ QIntersection: The intersection of 2 sets P and Q are all the elements that are both in P and Q. Denoted as; P∩QUnion: the union of P and Q are all the elements which are in P or Q. Denoted as; P∪QDisjoint/Mutually Exclusive: Two sets that have nothing in common. Ex: If P = {1, 3, 4} and Q = {2, 3, 5}, then

(a) Is P ⊆ Q? (b) What isP∩Q?

(c) What isP∪Q ?

Practice

Use sets M and N to answer the questions that follow.

M = {2, 3, 5, 7, 8, 9} N = {3, 4, 6, 9, 10}

(7) True or False: (a) 4∈M (b) 6 ∉ M(8) List the sets (a) M∩N (b) M∪N(9) Is (a) M ⊆ N (b) {9, 6, 3} ⊆ N

Bell-Ringer (Day 2)Using the following sets answer the questions that followA = { 1, 2, 3, 4, 5} B = { 6, 7, 8, 9} C = {4, 5} D = {4, 5, 6, 7}(1) List the elements in (a) A ∩ C (b) B ∩ D (c) A ∩ B (d) D ∩ C(e) A ∪ C (f) C ∪ B (g) C ∪ D

(2) Which sets are subsets?

Set Builder: is a mathematical notation that is used in set theory to describe the conditions that an element of that set must satisfy. Ex.) A = {x∨3<x ≤10 , x∈Z }

It reads as follows; The set of all x such that x is an integer between 3 and 10, including 10. The elements in the set are; A = { 4, 5, 6, 7, 8, 9, 10 }The number of elements in the set (Cardinality) is denoted; n(A) = 7

Practice

For each of the following sets given

(a) Write a sentence accurately describing the set.

(b) List the elements in the set, if possible.

(c) Give the cardinality of the set.

(3) B = {x∨−2≤x ≤4 , x∈Z }

(4) C = {x∨−2≤x<4 , x∈R }

(5) E = {x∨−1≤x ≤7 , x∈Z }

(6) D = {x∨−3≤ x≤7 , x∈N }

Day 3: Review and Quiz

Bell-Ringer (Day 4) Review of problem 2 from the quiz

For each of the following sets given

(a) Write a sentence accurately describing the set.

(b) List the elements in the set, if possible.

(c) Give the cardinality of the set.

(1) B = ¿

(2) B = {x∨−2≤x ≤7 , x∈Q }

Definition

The complement of a set A, denoted A', is the set of all elements of U which are not in A. U being the universal set.

Ex.) List the elements in set A' given;

U= {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 3, 5, 7, 8}

Corollaries (Just means follows from)

A∩A' = ∅ A ∪A' = Universal Set n(A) + n(A') = n(U)

Practice with complements

Write, in set builder notation, the set C' for the following given U = R and construct a line graph that clearly shows both C and C'.

(3) C = {x∨x<2 , x∈R }

(4) C ={x∨x ≥−4 , x∈R }

(5) C ={x∨−1≤x<3 , x∈R }

(6) C = {x∨7<x<11 , x∈ R }

(7) C = {x∨x ≤0 , x∈ R }

(8) C = {x∨x>6 , x∈R }

(9) C = {x∨−13≤ x≤0 , x∈R }

(10) C = {x∨−9<x≤0 , x∈R }

Warm up (Day 5)

Write, in set builder notation, the set C' for the following given U = Z and construct a line graph that clearly shows both C and C'.

(1) C = {x∨x<14 , x∈Z }

(2) C = {x∨x ≥−8 , x∈Z }

(3) C = {x∨−2<x<2 , x∈Z }

(4) C = {x∨0<x ≤5 , x∈Z }

New Practice Combining Everything

If U = {x∨−5≤ x≤5 , x∈Z } , A = {x∨1≤x ≤4 , x∈Z }

and B ={x∨−3≤ x<2, x∈Z }

list the elements in:

(5) A (6) B (7) A' (8) B'

(9) A∩ B (10) A ∪B (11) A' ∩ B (12) A' ∪ B'

Day 6 Review and Quiz

Quiz Review

Write, in set builder notation, the set C' for the following given U = R and also draw a line graph representing C and C’.

(1) C = {x∨x<11 , x∈R }

(2) C = {x∨−4<x ≤7 , x∈ R }

If U = {x∨−1<x ≤8 , x∈Z } , A = {x∨1≤x ≤6 , x∈Z }

and B ={x∨0≤ x<4 , x∈Z } list the elements in

(3) A (4) B (5) A' (6) B'

(7) A∩ B (8) A ∪B (9) A' ∩ B (10) A' ∩ B'

Day7

Warm Up

Suppose U = {Positive integers} P = { Multiples of 4 less than 50}

Q = {multiples of 6 less than 50}

list the elements in the following sets.

(1) P (2) Q (3) P∩Q (4) P∪Q

(5) Verify that; n (P∪Q )=n (P )+n (Q )−n(P∩Q)

Practice

Suppose U=Z and C={ y|−3≤ y ≤2 , y∈Z }

D= { y|−5≤ y<0 , y∈Z }

list the elements for problems 6 through 9

(6) C (7) D (8) C∩D (9) C∪D

(10) Verify that; n (C∪D )=n (C )+n (D )−n(C∩D)

Suppose U=Z+¿¿ A={Multiplesof 6 less than40 }

B= {Factors of 30 }

C={Prime ¿ ' s<42}

List the elements in the sets given for problems 11 through 17.

(11) A (12) B (13) C (14) A∩B

(15) B∩C (16) A∩B∩C (17) A∪B∪C

(18) Verify that; n ( A∪B∪C )=n ( A )+n (B )+n (C )−n ( A∩B )−n (B∩C )−n (A ∩C )+n(A∩B∩C )

Day 8

Do Now

Suppose U=Z+¿¿ P= {Factors of 18 }

Q={Multiplesof 4 between18∧50 }

R={Primes<20 }

List the elements in the sets given for numbers 1-7

(1) P (2) Q (3) R (4) P∩Q (5) Q∩R (6) P∩R

(7) P∪Q∪R

(8) Verify that:n (P∪Q∪R )=n (P )+n (Q )+n (R )−n (P∩Q )−n (Q∩R )−n (P∩R )+n(P∩Q∩R)

New Practice

If U=Z and A={ y|−5≤ y≤1 , y∈Z }

B= {y|0≤ y≤7 , y∈Z }

list the elements in the following sets if they are finite. If they are

infinite, write the set in set builder notation.

(9) A (10) B (11) A' (12) B' (12) A∩B

(13) A '∩B (14) B'∪A

Day 9

Do Now

Suppose U=Z A={x|−7≤ x≤3 , x∈Z }

B= {x|0≤x<7 , x∈Z }

list the elements in the following sets if they are finite. If they are infinite, write the set in set builder notation.

(1) A (2) B (3) A∩B (4) A'

(5) B' (6) A '∩B ' (7) A∩B' (8) A '∪B

Suppose U=Z A={x|5≤ x<9 , x∈Z }

B= {x|−1≤ x≤5 , x∈Z }

list the elements in the following sets if they are finite. If they are infinite, write the set in set builder notation.

(9) A (10) B (11) A∩B (12) A'

(13) B' (14) A '∩B ' (15) A∩B' (16) A '∪B

Suppose U=Z A={x|−2≤x<4 , x∈Z }

B= {x|−5≤ x<1 , x∈Z }

list the elements in the following sets if they are finite. If they are infinite, write the set in set builder notation.

(10) A (11) B (12) A∩B (13) A'

(14) B' (15) A '∩B (16) A∩B' (17) A '∪B

(18) A '∪B '

ANSWERS 10-18

Day 10

Review Quiz 5min

Copy Notes

Venn Diagram: is a visual representation of sets of objects, numbers, or things. It consists of a universal set U represented by a rectangle. Circles represent sets within the universal set.

Ex) The following illustrations represent many examples of Venn diagrams.

Practice

(1) Create a Venn diagram for each of the following, showing all elements in their proper placement;

(A) U={2 ,3 ,5 ,7 ,8 } A={2 ,7 ,8 }

(B) U={3 ,4 ,5 ,9 ,10 ,11 ,12 ,13 } A={5 ,9 ,12 }B={3 ,5 ,12,13 }

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

(D) U={10 ,11 ,12 ,13 ,14 ,15 ,16 ,30 } A={10,11,12}B={13 ,14 ,30 }

Day 11

Set A and its complement A’Two intersecting sets A and B and their intersection

Two intersecting sets A and B and their union

Do Now

(1) Given U={x|−5≤x<4 , x∈Z } for each pair of sets illustrate bothsets in a Venn diagram.

(a) A = {-3, -2, 1, 2} B = {-5, -4, -1, 0, 2, 3}

(b) A = {-3, -2, 0, 1, 2} B = {-2, 0, 1}

(c) A = {-5, -4, 0} B = {-3, -2, -1, 2}

(d) A = {1, 2, 3} B = {-2, 0, 1, 2, 3}

(2) Given U={x|−2<x≤7 , x∈Z } for each pair of sets illustrate both sets in a Venn diagram.

(a) A = {0, 1, 2, 7} B = {-1, 3, 4, 5, 6}

(b) A = {0, 1, 2} B = {-1, 0, 1, 2, 6, 7}

(c) A = {2, 3, 4, 5} B = {-1, 2, 3, 6}

(d) A = {-1, 0, 1, 2, 3, 4, 5, 6, 7} B = {-1, 0, 1, 2}

Venn Diagram shaded regions

(Ex) Given that sets A and B are intersecting sets draw a Venn diagram and shade

the appropriate region given.

(a) A'∪B' (b) (A'∩B ')'

Day 12

For each of the following sets given, draw a Venn diagram and shade the appropriate region representing each set given that A and B are intersecting sets.

(3) A '∪B (4) A∩B' (5) A∪B '

(6) (A∩B)' (7) (A∪B) ' (8) (A'∩B) '

Given A and B are disjoint sets, create a Venn diagram for each set and shade the region representing the set.

(9) A' (10) B' (11) A∪B (12) A '∩B

(13) A∪B ' (14) (A∩B)' (15) (A'∪B' )'

Day 13

More practice shading

Given A and B are intersecting sets, create a Venn diagram for each set and shade the region representing the set.

(16) (A∪B') ' (17) A '∩B ' (18) B'∪A ' (19) A '∩B

Given A, B, and C are intersecting sets, create a Venn diagram for each set and shade the region representing the set. Make sure to draw sets A, B, and C in all diagrams.

(20) A∩B∩C (21) B∩ A ' ∩C ' (22) C '∪B ' (23) A '∩B∩C

Day 14

Do now

Given A, B, and C are intersecting sets, create a Venn diagram for each set and shade the region representing the set. Make sure to draw sets A, B, and C in all diagrams.

(1) B∩C ' (2) (A∪B)∩C (3) (A∪C) ' (4) (B∩C)∪ A

Number Regions

Number in regions refers to Venn diagrams that only show or give us the number of elements in a particular region of the Venn diagram.

Example)

Notice that the numbers are in parenthesis. This is how IB notes that the number represents the number of elements in the set. (7) does not mean that the number 7 is an element of A∩B' but rather that there are 7 elements in the set A∩B' . Thus,

n(A) = 11 n(B) = 10 n(U) = 20

Practice

Answers:

More Practice

Use the following Venn Diagrm to find

(1) n(Q) (2) n(P∪Q) (3) n(Q ') (4) n(U )

Find the value aof

(5) n (U )=29 (6) n (U )=31

Day 15

(2a) (a+4)(a)

(a-5)

P Q

U

(1) Find the number of elements in each set. Use the following Venn diagram.

(a)A ' (b) B' (c) (A∩B)' (d) A∩B' (e) B' ∩A '

New Type of Venn diagram Problem

(2) If A and B are 2 intersecting sets and n (U )=30 ,n ( A )=14 , n (B )=17

and n ( A∩B )=6, determine;

(a) n(A∪B) (b) n(A ,but not B)

(3) If A and B are 2 intersecting sets and n (U )=26 , n (A )=11 , n (B )=12

and n ( A∩B )=8 determine;

(a) n(A∪B) (b) n(B ,but not A)

(4) If A and B are 2 intersecting sets and n (U )=32, n ( A )=13 ,n ( A∩B )=5

and n ( A∪B )=26 determine;

(a) n(B) (b) n( (A∪B )' )

(5) If A and B are 2 intersecting sets and n (U )=50 ,n ( A )=30 , n(B)=25

(9) (4)(2)))

(6)U

and n ( A∪B )=48 determine;

(a) n(A∩B) (b) n(B ,but not A)