· Web view2015-09-24 · Unit 1 Sets. Definitions. (Day 1). Sets: A collection of numbers or...
Transcript of · Web view2015-09-24 · Unit 1 Sets. Definitions. (Day 1). Sets: A collection of numbers or...
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)