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A set is referred to as the well-defined collection of objects known as elements or members of the set. Name and represent sets use capital letters. In set theory, the elements of a set can be anything: people, letters of the alphabet, numbers, shapes, variables, and so on. The types of sets include finite set, infinite set, subset, null set, power set, equal set, equivalent set, proper, improper sets and many more.
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Key Takeaways: Sets, finite set, infinite set, the cardinality of a set, definite set, disjoint set
Sets Definition
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A set is a predefined grouping of objects. A definite set is one that contains a fixed number of objects. Indefinite sets, on the other hand, are made up of an indefinite number of elements.
Examples
Finite set: {1,2,3,4}.
Infinite set: the set of all the integer numbers.
Symbols Used in Sets
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| Symbols | Meaning |
|---|---|
| N | Natural set of numbers |
| Z | Integer set of numbers |
| Q | Rational set of numbers |
| R | Set of real numbers |
| Z+ | A positive set of integers |
| Q+ | A positive set of rational numbers |
| R+ | A positive set of real numbers |
A U B is used to represent the unification or addition of two sets A and B. Finding the union of two sets yields a set that contains all of the elements in both A and B.
If you want to find the common elements between two sets A and B, you must first find the set A intersection B or the A inverted U B.
Sets can be added and subtracted from one another.
You can also find A bar, which will show you all the elements that are not included in the set. This is referred to as the complement of the set.
Cardinality of a Set
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A set's cardinality can be defined as the number of elements it contains. It could be from 0 to infinity.
As an example,
Consider the set A = 1,2,3,4 as an example.
Set A's cardinality is represented as n(A), which is 4 because A contains 4 elements.
Also read: Union of Sets
Set Formula
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Following are some basic formulas from the set theory:
(A) Group of two sets A, B
If A and P are overlapping set, n(A∪P)=n(A)+n(P)–n(A∩P)
If A and B are disjoint set, n(A∪B)=n(A)+n(B)
n(U)=n(A)+n(B)–n(A∩B)+n((A∪B)c)
n(A∪B)=n(A−B)+n(B−A)+n(A∩B)
n(A−B)=n(A∩B)−n(B)
n(A−B)=n(A)−n(A∩B)
n(Ac)=n(U)−n(A)
(B) For a set of three sets P, Q, and C
n(P∪Q∪C)=n(P)+n(Q)+n(C)−n(P∩Q)−n(Q∩C)−n(C∩P)+n(P∩Q∩C)
Things to Remember
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- Set theory is a branch of mathematics concerned with the properties of well-defined collections of objects that may or may not be mathematical in nature, such as numbers or functions.
- The theory of sets as it has evolved George Cantor is now used in all branches of mathematics. ''A set is a well-defined set of object categories of our impression or thought that must be brought into existence as a whole,`` he explains.
- Sets with a finite/countable number of members are called finite sets. If the elements of this set have a finite number of members, the process will run out of elements to list.
- If a set is not finite, it is referred to as an infinite set because the number of elements in that set cannot be counted.
Sample Questions
Ques. In a class of 100 students, 35 enjoy drawing and 45 enjoy music. Both are rated ten. Determine how many of them like both of them or neither of them. (3 marks)
Ans. Total number of students, n(μ) = 100
Number of drawing students, n(d) = 35
Number of music students, n(m) = 45
The number of students who enjoy both, n(d∩m) = 10
Number of students who like one or both of them,
n(dá´?m) = n(d) + n(m) – n(d∩m)→ 45+35-10 = 70
Number of students who dislike both = n(μ) – n(dá´?m) = 100 – 70 = 30
Ques. There are 200 students in a school, 65 of whom enjoy drawing and 85 who enjoy music. 25 approves of both. Determine how many of them like both of them or neither of them. (3 marks)
Ans. Total number of children, n(μ) = 200
the number of children who like to draw, n(d) = 65
The number of children who like music,n(m) = 85
Number of students who like both, n(d∩m) = 25
Number of students who like one or both of them,
n(dá´?m) = n(d) + n(m) – n(d∩m)→ 65 + 85 - 25 = 125
Number of students who like neither = n(μ) – n(dá´?m) = 200 – 150 = 50.
Ques. If A and B are two finite sets with n(A) = 20, n(B) = 28, and n(A B) = 36. (2 marks)
Ans. Using the equation n(A B) = n(A) + n(B) - n(A B).
then n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 20 + 28 - 36
= 48 - 36
= 12
Ques. If n(A - B) = 18, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B). (3 marks)
Ans.
Using the formula n(A∪B) = n(A - B) + n(A ∩ B) + n(B - A)
70 = 18 + 25 + n(B - A)
70 = 43 + n(B - A)
n(B - A) = 70 - 43
n(B - A) = 27
Now n(B) = n(A ∩ B) + n(B - A)
= 25 + 27
= 52
Ques. In a group of 60 people, 27 prefer cold beverages and 42 prefer hot beverages, and each prefers at least one of the two. How many people like coffee and tea? (3 marks)
Ans. Let A represent a group of people who enjoy drinking cold beverages.
B = A group of people who enjoy hot beverages.
Given
(A ∪ B) = 60 n(A) = 27 n(B) = 42 then;
n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 27 + 42 - 60
= 69 - 60 = 9
= 9
Therefore, 9 people like both tea and coffee.
Ques. The art class has 35 students and dance class has 57 students. Determine the number of students enrolled in art or dance classes.
a)When two classes meet at different times and a total of 12 students participate in both activities.
b)When two classes are scheduled to meet at the same time. (4 marks)
Ans. n(A) = 35, n(B) = 57, n(A ∩ B) = 12
(Assume A is a group of art students.)
B denotes the group of students in dance class.)
(i) When 2 classes meet at different hours n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 35 + 57 - 12
= 92 - 12
= 80
(ii) When two classes meet at the same hour, A∩B = ∅ n (A ∪ B) = n(A) + n(B) - n(A ∩ B)
=n(A)+n(B)
= 35 + 57
= 92
Ques. Art class has 35 students and dance class has 57 students. Determine the number of students enrolled in art or dance classes.
a)When two classes meet at different times and a total of 12 students participate in both activities.
b)When two classes meet at the same time. (5 marks)
Ans. Assume A is a group of people who speak English.
B denotes the group of people who speak French.
A-B represents the group of people who only speak English and not French.
B - A is a group of people who speak French rather than English.
A B is a group of people who can communicate in both French and English.
Given,
n(A) = 72 n(B) = 43 n(A ∪ B) = 100
Now, n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 72 + 43 - 100
= 115 - 100
= 15
Therefore, Number of persons who speak both French and English = 15
n(A) = n(A - B) + n(A ∩ B)
⇒ n(A - B) = n(A) - n(A ∩ B)
= 72 - 15
= 57
and n(B - A) = n(B) - n(A ∩ B)
= 43 - 15
= 28
As a result, the number of people who only speak English is 57.
The number of people who only speak French is 28.
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