Set theory

I. CONCEPT
Strictly speaking, the "Set" is considered as a non-concept
defined, getting accustomed to using as synonyms for sets the words:
«colección», «reunión», «agregado», etc.
That is why we can affirm that the word 'set' gives us the idea
of grouping homogeneous objects of real or abstract possibilities.
The members belonging to the group are called 'ELEMENTS'
from the set.

II. NOTATION
"A" is the set whose elements are the letters of the alphabet.
A = {a, b, c, .........., z}

III.CARDINAL OF A SET (n)

The cardinality of a set is the number of elements it has.
a set.
n(A) It reads: 'Number of elements in set A'
EXAMPLE:
A = {2; 4; 6; 8; 10} n(A) = 5
B = {1; 1; 2; 2} n(B) = 2
C = {{2; 3}; {7; 8}} n(C) = 2

IV. RELATION OF BELONGING

It is the one that relates every single element of a set,
said set.
Element∈ Set
Examples:
* A = {5, 10, 15, 20, 25} 5∈ A: '5 belongs to set A'
Also: 10∈ A ; 20∈ A ; 21∉ A.
* B = {2; 3; {4}; 5} 2∈ B ; 3∈ B ; 5∈ B ; 4∉ B ; {4}∈ B
V. DETERMINATION OF SETS:
1. For understanding or constructively. When defining the set
stating one or more common properties characterizes the elements
of that set.
2. By Extension or in Tabular Form: It is when they are listed one by one.
all or some of the elements of the set.
Example:
A) Determine the set of vowels.
B) Determine the set of odd numbers less than 16.
SOLUTION
By Extension: For Understanding:
A = [a, e, i, o, u] A = [x / x is a vowel]
B = [1, 3, 5, 7, 9, 13, 15] B = [x/x is an odd number, x < 16]

OBSERVATION:
x/x is read as: 'x is an element of the set such that x'.

VI. RELATIONS BETWEEN SETS

1. INCLUSION(⊂It is said that a set "A" is included in a set
All elements of 'A' belong to 'B'.
Example:
Si: A = {a, b, {c}} y B = {a, b, {c}, d}
A⊂ B Also:
A is included in B {a, b}⊂ A ; * {c, d}⊄ B
A is part of B * {b, {c} }⊂ A ; * {A} ⊄A
A is contained in B
A is a subset of B
OBSERVATION:
Conventionally, the empty set is considered to beφ) is included in
whole set.
φ ⊂ Aφ ⊂ B
SUBSET:
Let the setAbe a subset ofA, every set included in setAis a subset ofA.
 Example:
Yes: A = {a, b, c}
Subsets of A:
{a}; {b}; {c}; {a, b}; {a, c}; {b, c}; {a, b, c};φ
Then "A" has 8 subsets.
Number of Subsets of A = 2n(A)

2. PROPER SUBSET. Given a set "A", a proper subset

The set of "A" is all those subsets of "A", except the one that is equal to it.

Number of Proper Subsets of A = 2n(A)- 1

EQUALITY OF SETS. Two sets are said to be equal if

they have the same elements.
A = B⇔ A⊂ B by B⊂ A
EXAMPLE:
Yes:
A = {1, 3, 5, 7, 9} A= B
B = {x∈ N / X odd < 10

EXAMPLE:
If A and B are equal sets, find X+Y
Yes: Invalid input formatx- 1; 27} and B = {3y-1; 31}
RESOLUTION
The elements of A are the same as those of set B; then they
deduce
2x31 * 27 = 3y-1
2x = 32 33= 3y-1
x=5 3=y-1
y=4 ∴x+y=9

3. DISJOINT SETS. Two sets are disjoint when they do not have
common elements.
EXAMPLE:
P = {2; 4; 6; 8} ; I = {1; 3; 5; 7}
VII. TYPES OF SETS BYTHE NUMBER OF ELEMENTS:
1.UNIT SET. It is the set that consists of only one
element.
S = {X∈ N / 3 < X < 5 X=4
S = {4} n (S) = 1
EXAMPLE:
Yes: A = {a2- 6; a + b; 10} is Unit.
Find: a x b; if aN
RESOLUTION
The 3 elements are the same (equal).
* a2- 6 =10 * a + b = 10
a2=16 a x b = 24
a=4
4
6
2.EMPTY SET(φIt is the set that has no elements;
it is also called the null set. By convention, it is agreed that the
The empty set is a subset of any other set.φ A).
R = {x∈ N / 5 < x < 6 there is no value for 'x'
R = { } =φ n(R) = 0

3.FINITE SET. It is a set with a limited number of elements.

It can be determined by extension.
F = {x∈ Z / 3 < x < 12 F = {4; 5; 6; .......; 11}
4.INFINITE SET. It is a set that has an unlimited quantity.
of elements:
A = {x / x∈ Z; x > 0}
A= {1, 2, 3, 4, ......} n(A) =∞

VIII. OTHER CONCEPTS:

1.UNIVERSAL SET (U). It is a reference set; for analysis
from a particular situation, it is chosen arbitrarily.
Example:
A = {x / x is a hen}
 You can take:
U = {x / x is a bird} or U = {x / x is a vertebrate}
2.POWER SET[P(A)]. Given a set A, the power set
The power set P(A) is the set that is formed by all the subsets of A.
Yes: A= {a, b, c}
P(A) = {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, φ}
Then: P(A) has 8 elements
n [P(A)] = 2n(A)
Example:
How many elements does the power set of C have?
C = {2, 4, 6, 8, 10}
Resolution:
As n(C) = 5 n [P(C)] = 25= 32

IX. VENN–EULER DIAGRAMS

They are closed flat regions, circular, rectangular, etc. That we
will allow the graphical representation of sets.
Example:
Given the sets:
A = {2, 4, 6} ; B = {3, 4, 5} ; C {7, 8, 9} ; U = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10}

A B C
7
2 3
4

6 5 8 9

X. CARROLL DIAGRAM
With greater utility for distinct sets.
APPLICATION:
In a classroom of 90 students, 35 are women, 62 are athletes, and 12 are
Women who are not athletes. How many men are not athletes?
Resolution:
M = 35 H = 55
Dep. = 62
No Dep.= 28 12 X No Athletes:
90 12 + X = 28
X = 16

SOLUTION PROBLEMS
1. Given: C = {m + 3/ m∈ Z; m2< 9}
Calculate the sum of elements in the set C

Yes: m∈ z y m2< 9
↓ ↓
-2 4
-1 1
0 0
1 1
2 4
If: Elements: (m + 3) C = {1; 2; 3; 4; 5}
∴ Σ elementos: 15

2. There are two sets where one is included in the other; the difference of
The cardinality of its power sets is 112. Indicate the number of
elements that the set includes the other.

Sets A and B (B⊂ A)

Yes: xy
Given: n [P(a)] – n [P (B)] = 112
 
2x + n 2x–112 = 16.7

2x. (2n- 1) = 24. (23– 1)

Then: X = 4 and n = 3
∴ n(B) = 4 y n (A) = 4 + 3 = 7

3. Yes: A = {x/x∈ Z ^ 10 < x < 20

B = {y + 5 / y∈ Z ( y + 15)∈ A}
What is the sum of the elements of B?

Set A, determined by extension, is:

A = {11; 12; 13; 14; 15; 16; 17; 18; 19}
In set B, like ( y +15)∈ A

10 < y +15 < 20 -5 < y <5

y = 0, 1, 2, 3, 4 why and∈ Z y = 0, 1, 4, 9, 16
Then: B = {5; 6; 9; 14; 21}
∴ Sum of elements of B = 55
 PROPOSED PROBLEMS
Given the set A lady goes out for a walk every day.
days with two or more of their puppies.
B = {14; {2};φ ; {7; 15}}
With great care, he tried to carry
{2} ⊂ B {14}∈ P(B)
every day a different group. Yes
{7; 15}∈ Bφ ∈ Β
In total, he has 10 puppies. What about
φ⊂Β {14; φ} ⊂B
how many days will it have to be
14⊂ B 14∉ B
necessarily lead to a group
{{2}; 14}∈ P(B)
Howmanypropositionsarethere?
repeated?
A) 10 B)11 C) 12
false?
D) 13 E) 14
A) 3 B) 1 C) 5
D) 4 E) 6 8. given the sets:
2. Determine the extension of the 2a+ 1a
A= { / ∈N∧ 1≤ a≤ 9}
next set: 3 2
A = {(3x - 3) / x∈N∧ 0≤ x≤ 4}
2b−1/ b∈ N; 2 < b≤ 6}
A) {0; 1; 2; 3} B) {1; 2; 3} B={
C) {0; 3; 6} D) {0; 3; 6; 9} 3
E) { -3; 0; 3; 6} Determine: E = [n(B)]n(A)+ n(A).
3. If A = {(x2+ 4) / x∈Z∧ -4 < x < 6 A) 270 B) 120 C) 200
Find n(A) D) 180 E) 260
A) 4 B) 5 C) 6
D) 7 E) 8 9. Given the equal sets:

4. If B = {(x + 1) / x∈N∧ 3x < x + 14 A = { a+2; a+1}, B = {b+1; c +1},

Give as a response the cardinal of C = {7 - a; 8 - a} and D = {b + 2; d + 3}. Find
B. "a+b+c+d" if additionally b≠ d
A) 4 B) 5 C) 6 A) 10 B)11 C) 12
D) 7 E) 8 D) 13 E) 14
5. Calculate (b - a) if E is a set 10. What is the sum of the elements?
unitary. E = {4a+1; 2b+a; 3a+4} of set A? yes:
A) 1 B) 2 C) 3
D) 4 E) 5 A = {2x / (3x+1)}∈ N y 4 < x < 8
A) 36 B) 165 C) 116
6. Given the sets A = {1; 2; 3; 4; D) 160 E) 132
5; 6} and B = { 0; 1; 4; 6; 7; 8; 9}. Let it be
m the number of subsets not 11. Make 2 comparable sets
empty sets of A that are disjoint with whose cardinals are differentiated
By "n" the analogous from B to A. Find in 3. also the difference between
m+n the cardinals of their sets
A) 7 B) 7 C) 22 power is 112. Indicate the number
D) 24 E) 26
 of terms that the set has 17. Given the unit set:
which includes the other. 3a - 3b + 2; a + b; 14
A) 5 B) 4 C) 7 Determine the number of
D) 6 E) 9 proper subsets of
B = {a; 2a; b; 2b - 1}
12. How many ternary subsets
A) 7 B) 15 C) 31
it has a set whose cardinal
Is it 12?
D) 63 E) 8
A) 220 B) 224 C) 218 Give the sum of the elements of
D) 216 E) 200 A = {2x/ x∈ N; 10 < 3x + 2 < 18
A) 19 B) 18 C) 24
13.Given the sets:
A = {x / x∈ From∧ -3≤ x≤ 10} D) 26 E) 23
B = x / x∈ N∧ y = 2x - 3∧ y∈ A} 5x+ 2
C = {x / x∈ B∧ 4 < x + 3 < 7 19. If P = {x2-1/-6< <6;x∈ Z+}
5
find the sum of the elements
Deetrmniethenumberof
of set C
subsets.
A) 2 B) 3 C) 5
A) 16 B) 64 C) 32
D) 8 E) 11
D) 8 E) 128
The set A has 14 subsets.
3x−1 ∈Z/1<
ternary more than binary. 20.si:Q = { x< 3; x∈ N}
How many unit sets are there? 4
A? find the sum of elements of Q
A) 5 B) 6 C) 7 A) 35 B) 15 C) 12
D) 8 E) 9 D) 11 E) 7
15. Find the sum of the elements of 21. Set A = {m + n; 4} a set
a−1 unitary and B = {2m-2n; m+n} has
M = {a / ∈ N; a < 73 a cardinal equal to 1. Find the value
2 of m/n.
A) 111 B) 113 C) 110 A) 3 B) 4 C) 6
D) 115 E) 116 D) 5 E) 0
16.given the set: 22. Find the sum of the elements of:
A = {4; 8;φ; {4}; {2; 7}; {φ}
} Determine how many of the B={
n 2−16
/ n∈ Z; 0 < n≤ 5}
the following propositions are n− 4
true
2; 7∈ A 4∈ A
Invalid input.φ } ⊂ Α {4; 8} ⊂ A A) 35 B) 36 C) 27
{2; 7} ⊂ A {{ φ}} ⊂ Α D) 0 E) 25
φ∈A {{4}; {2;7}}⊂A
A) 5 B) 4 C) 7
D) 3 E) 6
23. Let the sets be: 27. A "chubby" enters a
A = {2x / x∈ Z; 0 < x < 6 restaurant where they serve 6
x+ 4 different dishes and think 'me
B={ / x∈A} I like everyone but I have to take
2
at least 2 dishes and 5 like
2y+1 maximum" How many ways
C={ ∈Z / y∈B
3 Can you choose the "chubby" one?
Find the cardinality of P(C) A) 64 B) 56 C) 32
A) 4 B) 8 C) 9 D) 26 E) 120
D) 16 E) 32 28. If A and B are singleton sets
24.given the set: How many elements does C have?
A = {x + 4 / x∈ N; x2< 16} a + 2b; 17
calculate the sum of the elements C = {x / x∈ N;a≤ x≤ b}
from A. A) 5 B) 6 C) 7
A) 10 B) 16 C) 19 D) 4 E) 2
D) 27 E) 28 29. Consider the sets:
25. How many proper subsets A = {x / x∈ Z;0≤ x< 10} y
does that set have 35 B = {2n∈ A / (n/3)∈
How many subsets does it have?
ternary subsets?
set P(B)?
A) 127 B) 63 C) 31
A) 16 B) 4 C) 8
D) 1023 E) 511
D) 32 E) 64
Given the set:
30.Given the sets:
x x 2−1 A = {x / x∈ Z;8≤ x≤ 19}
A={ / x∈ Z; -3 < ≤ 1}
x−1 x+ 1 B = {y + 4 / y∈ N;(2 y - 1)∈ A}
What is the sum of the elements? Find the sum of the elements of
from A? set B
A) 2 B) 3 C) 4 A) 350 B) 379 C) 129
D) 5 E) 6 D) 252 E) 341

KEYS
01. A 02. E 03. C 04. D 05. B 06. C 07. E 08. E 09. B 10. E
11.C 12. A 13. C 14. C 15. D 16. E 17. A 18. C 19. B 20. E
21. A 22. C 23. A 24. E 25. A 26. A 27. B 28. A 29. A 30. B