SETS I. SETS AND MEMBERS OF SET Example:
A = { 1, 2, 3, 4, 5, 6} The members of set A are 1, 2, 3, 4, 5, and 6, there for we can
write: 1 2 3 4 5 6
‘ ’
∈ ∈ ∈ ∈ ∈ ∈
A A A A A A
and
7∉A 9∉A 12 ∉ A 16 ∉ A 20 ∉ A 35 ∉ A
means element
and
‘ ’
means not
element II. EXPRESSING SETS Three ways to expressing sets, there are: 1. By expressing sets in words Example: B = { prime less than ten } 2. By roster way Example: C = { 2, 3, 5, 7 } 3. By forming notation Example: D = { x x < 10, x ∈ prime }
III. CARDINALITY OF SETS Look at the following sets: E = { a, b, c, d, e } F = { 1, 3, 5, 7, … , 15 } G = { 3, 6, 9, 12, … } H = { …, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 } - Set E has 5 elements, elements, we can write n (E) = 5 - Set F has 8 elements, elements, we can write n (F) = 8 For sets E and F we can count the number of the elements of set, thus sets E and F are called countable sets or finite sets. Now for sets G and H, we are not able to count the number of elements uncountable le sets or of both both sets sets.. Thes These e kind kinds s of sets sets are are call called ed uncountab infinite sets.
WORK SHEET 1. Look at the following sets: K = { 5, 10, 15, 20, 25, 30 } L = { 1, 2, 3, 4, 6, 8, 12, 24 } M = { 4, 8, 12, … , 48 } Answer the questions below by fill in the blank. a. The elements of K are …, …, …, …, …, and …. Fill the blank by ‘ ’ or ‘ ’ Thus 5…K 20 … K 25 … K 6 … K 14 … K 50 … K b. The elements of L are …, …, …, …, …, …, …, and …. Fill the blank by ‘ ’ or ‘ ’ Thus 2…L 4…L 12 … L 21 … L 28 … L 32 … L c. The elements of M are …, …, …, …, …, …, …, …, …, …, …, and …. Fill the blank by ‘ ’ or ‘ ’ Thus 16 … M 36 … M 40 … M 52 … M 44 … M 72 … M 2. Expressing the following sets using three ways. a. The set of the first five natural numbers. Answer: - By expressing sets in word: { the ………………………………… ………} - By roster way: { …, …, …, …, … } - Forming Forming notation: { x ………………………………… …..} b. The set of the factor of 18 Answer: - By expressing sets in word: { the ………………………………… ………} - By roster way: { …, …, …, …, …, … } - Forming Forming notation: { x ………………………………… …..} c. The set of the letter on word “ MATEMATIKA” Answer: - By expressing sets in word: { the ………………………………… ………} - By roster way: { …, …, …, …, …, … }
- Forming Forming notation:
{ x …………………………………
…..} 3. Expressing the following sets by the others ways. a. N = { x 2 < x < 7 , x natural numbers} Expressing by word : { ……………………between ……………………between ………………… } b. P = { th the e odd number between 6 and 36 } Expressing by roster roster way: way: { …, …, …, …, …, …, …, …, …, …, …, …, …, …, … } c. Q = { 4, 6, 8, 10, 12, 12, 14, 16, 18 18 } Expressing by forming notation: { x ……………………………………..}
4. Look at the following sets. R = { 1, 2, 3, …, 10 } S = { 10, 10, 20, 30, 30, 40, … } T = { the factor of 56 } V = { x 3 < x < 18 , x even number} W = { the odd prime numbers } a. The countable sets are …, …, and …. b. The uncountable sets are … and …. c. Counting every sets and thus: n (R) = …. n (S) = …. n (T) = …. n (V) = …. n (W) = ….
IV. VENN DIAGRAM A. UNIVERSAL SETS ( U ) Example: Given A = { 2, 3, 5, 7, 11 } The possible universal sets of A are: 1. U = { 1, 2, 3, …, 15 } 2. U = { the first first ten prime prime number number } 3. U = { 1, 1, 2, 2, 3, 3, … , 20 20 } 4. U = { natur natural al numb number ers s} 5. U = { 2, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 29, 29, 31, 37, 37, 41 } 6. etc.
B. VENN DIAGRAM Examples: 1. Given U = { 1, 2, 2, 3, 4, 5, 6, 6, 7, 8 }, and A = { 2, 3, 5, 7 } The Venn Venn diagram of those sets is:
U .1
.2 A
.4
.3 .5
.6
.7
.8
2. Given U = { b, e, r, k, a, h }, A = { k, e, e, r, a, }, and B = { r, e, b, a, h } The Venn Venn diagram of those sets is:
U
A .k
B .b .h
.e .r .a
WORK SHEET Completing the dot on the following Venn Venn diagram based on the each sets. 1. U = { 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10 }, and and A = { 2, 4, 6, 8, 8, 10 }
U
.
.
. .
A
. .
. . 2. U = { a, b, c, d, e, f, g, h }, and B = { a, c, e, g }
U
B
. .
. .
. . . .
3. U = { 1, 2, 3, … , 10 10 }, C = { 1, 3, 5, 5, 7, 9 }, and D = { 3, 3, 6, 9 }
U
C
.
. .
. .
.
.
D
. .
.
4. U = { 1, 2, 3, … ,10 }, }, E = { 1, 2, 2, 4, 6, 6, 7, 8}, F = { 2, 2, 3, 5, 5, 6, 9 }, and and G = { 2, 4, 8, 9, 10 }
U E
. . .
. . .
G
F
. .
. .
C. EMPTY SETS AND SUBSETS 1. EMPTY SET ( ) Given: K = { the natural number between 3 and 4 } L = { the prime number less than 2 }
Set K and set L are called empty set , because both sets are not have elements. elements.
2. SUBSET Let P and Q be given sets. a. Set P is a subset of Q, written P ⊂ Q , if every elements of P is also an elements of Q. Q , if there is an element of P b. Set P is not subset of Q, written P is not an element of Q. Example: A = { 1, 2, 3, 4, 5, 6 }, B = { 1, 3, 3, 5 }, and C = { 3, 6, 9} Thus B
⊂ A, because every elements of B are also the
elements of A.
C
A, because 9 is element of C but is not
element of A.
WORK SHEET 1. Write down five examples of an empty set. a. ……………………………………………………………………………………………… …………… b. ……………………………………………………………………………………………… …………… c. ……………………………………………………………………………………………… …………… d. ……………………………………………………………………………………………… …………… e. ……………………………………………………………………………………………… ……………
2. Given set A = { a, b, c, d, e }, B = { a, d, e }, and C = { i, d, a } a. Is every element of A also an element of B? Thus, A … B ( fill ‘ ⊂ ’ or ‘ ⊄ ’ ) b. Is every element of A also an element of C? Thus, A … C c. Is every element of B also an element of A? Thus, B … A d. Is every element of B also an element of C? Thus, B … C e. Is every element of C also an element of A? Thus, C … A f. Is every element of C also a lso an element of B? Thus, C … B 3. Given set D = { 4, 6, 10, 16 }, determine and count the number of: a. Subset of D are not have element.
b. Subset of D are have one element. c. Subset of D are have two elements. d. Subset of D are have three elements. elements. e. Subset of D are have four elements. Answers: a. ∅ → 1 subset b. { … }, { … }, { … }, and { … } → … subset c. { …, … }, { …, … }, { …, … }, { …, … }, { …, … }, and { …, … } → … subset d. { …, …, … }, { …, …, … }, { …, …, … }, and { …, …, … } → … subset …,, …, …, … } → … subset e. { … 4. Determine the number of subset of a set. N Set The Subset The o number of number the of subset element of set 1 0 1 ∅ ∅ 2 {a} 1 ∅, … {…} ∅ 3 { a, b } … … { … }, { … } { …, … } ∅ 4 { a, b, … … { … }, { … }, { … } c} { …, … }, { …, … }, { …, … } { …, …, … } ∅ { … }, { … }, { … }, { … } 5 { a, b, c, … … { …, … }, { …, … }, { …, … },{ …, d} …} { …, …,… …,… }, { …, …, …, … }, { …, …, …, … }, { …, …, … } { …, …, …, … }
Based on the table above. If the number of set is
n, so the number of the subset of the set s et is
(…)
…
5. Given Given A = { 1, 1, 2, 3 }, B = { 1, 2, 3 3,, 4, 5 }, and C = { 1, 2 2,, 3, … , 10 } True or false : a. A ⊂ B (…)
b. B ⊂ C (…) c. A ⊂ C (…) 6. Given Given K = { a, a, b, c, d, e, f, g, h }, L = { a, e }, and M = { a, b, c, d, e, e, f } True or false : a. L ⊂ M (…) b. M ⊂ K (…) c. L ⊂ K (…) Based on no. 5 and no. 6 above.
If A ⊂ … and B ⊂ …, then A ⊂ … V. THE OPERATION OF TWO SETS A. INTERSECTION SETS( ) The intersection of set A and set B is a set formed by the condition that all elements of A also become elements of B. Notation:
A
B={x
WORK
x
A and x
B}
SHEET
1. a. Given A = { 1, 2, 3, 4, 5, 6 } and B = { 2, 4, 6 ,8 ,10 }. Based on both sets, the same element of those are …, …, and …. Those elements are the intersection of set A and set B, or
A
B = { …, …, … } B is The Venn diagram of A U
A
B
Shaded the part the elements of A ∩ B
b. Given: C = { a, b, c } and D = { a, b, c, d, e }. Based on both sets, every element in C also an element of D, so C … D, the same element of those are …, …, and …. D = { …, …, … } = …. and C
The Venn diagram of C
D is
U
D
Shaded the part of C ∩ D
C
c. Given: P = { 2, 3, 5, 7, 11, 13 } and Q = { the first six of prime number } Based on both sets, every element element of P are ………… ………… to Q or P … Q, the same element of those are …, …, …, …, …, …, and …. Q = { …, …, …, …, …, … } = … = … and P The Venn diagram of P
U
Q is
P=Q
Shaded the part of P ∩ Q
d. Given: K = { c, a, b, e } and L = { p, u, r, i } Based on both sets, every element of K are not ……….. to L, Are they have the same element? and K L = …. The Venn diagram of K
L is
U K
L
e. Look at the Venn diagram below
U
M
.4 .2 .6 .8 .3 . 10 .5
N
.7
.1
. 11 .9
Based on the Venn Venn diagram above, abo ve, determine: • U = {…, …, …, …, …, …, …, …, …, …, … } • M = {…, …, …, …, …, …, … } • N = {…, …, …, …, … } • M N = {…, …, …}
B. UNION SETS (
)
The union of set A and set B is a set formed by the condition that all elements of A or B. Notation:
A
B={x
WORK
x
A or x
B}
SHEET
1. a. Given A = { 1, 2, 3, 4, 5, 6 } and B = { 2, 4, 6 ,8 ,10 }. Based on both sets, the union element of those are …, …, …, …, …, …, …, and …. Those elements are the union of set A and set B, or
A
B = { …, …, …, …, …, …, …, … } B is The Venn diagram of A U
A
B
Shaded the part of A ∪ B
b. Given: C = { a, b, c } and D = { a, b, c, d, e }. Based on both sets, every element in C also an element of D, so C … D, the union element of those are …, …, …, … and ….
and C
D = { …, …, …, …, … } = …. D is The Venn diagram of C U
D
Shaded the part of C ∪ D
C
c. Given: P = { 2, 3, 5, 7, 11, 13 } and Q = { the first six of prime number } Based on both sets, every element element of P are ………… ………… to Q or P … Q, the union element element of those are are …, …, …, …, …, …, and …. Q = { …, …, …, …, …, … } = … = … and P The Venn diagram of P
U
Q is
P=Q
Shaded the part of P ∪ Q
d. Given: K = { c, a, b, e } and L = { p, u, r, i } Based on both sets, every element of K are not ……….. to L, The union element of those are …, …, …, …, …, …, …, and …. L = {…, …, …, …, …, …, …, … } and K The Venn diagram of K
L is
U K
L
Shaded the part of K ∪ L
e. Look at the Venn diagram below
U
M
.4 .2 .6 .8 .3 . 10 .5
N
.7
.1
. 11 .9
Based on the Venn Venn diagram above, abo ve, determine: • U = {…, …, …, …, …, …, …, …, …, …, … }
•
M
N = {…, …, …, …, …, …, …, …, … }
III. COMPLEMENT AND SET DIFFERENCE 1. COMPLEMENT OF A SET Given set A and its universal set is U. Then the complement of A is:
A ={x
x
U and x
A}
2. THE DIFFERENCE BETWEEN TWO SETS Given set A and B. Then the difference is:
A–B={x B–A= x
x x
WORK
A and x B and x
B} A
SHEET
A. 1. Given: U = { 1, 2, 3, …, 10 } and A = { 1, 2, 3 , 6 }. Based on both sets, the element of U are not the element of A are …, …, …, …, …, and …. Those element is called the Complement of A, A, or A’ = { …, …, …, …, …, … } The Venn diagram of A’ is
U
Shaded the part of A�
A
2. Given: U = { a, b, c, d, e ,f, g, h } and B = { b, e, d, a, h }. Based on both sets, the element of U are not the element of B are …, …, and …. Those element is called the Complement of B, B, or B’ = { …, …, … } The Venn diagram of B’ is:
U
B
Shaded the part of B �
B. Given: A = { 2, 4, 6, 8, 10, 12, 14 } and B = { 3, 6, 9, 12, 15, 18 } Based on both sets: 1. The elements of A that are not included in B are …, …, …, …, and …, Those are the element of A – B or
A – B = { …, …, …, …, … } The Venn diagram of A of A – B is:
U
A
B
Shaded the part of A �B
2. The element of B that are not included in A are …, …, …, and …, Those are the element of B – A or
B – A = { …, …, … } The Venn diagram of B of B – A is:
U
A
B
Shaded the part of A �B
