Set theory is a branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions.
Types of Set
The sets are further categorised into different types, based on elements or types of elements. These different types of sets in basic set theory are:
- Finite set: The number of elements is finite
- Infinite set: The number of elements are infinite
- Empty set: It has no elements
- Singleton set: It has one only element
- Equal set: Two sets are equal if they have same elements
- Equivalent set: Two sets are equivalent if they have same number of elements
- Power set: A set of every possible subset.
- Universal set: Any set that contains all the sets under consideration.
- Subset: When all the elements of set A belong to set B, then A is subset of B
On this page you will find few practice question in Mathematics (Set Theory) which will enhance your preparation for the forthcoming GCE Exam.
TOPIC: SET THEORY
Watch the video tutorial below, for detailed explanation of Set Theory, then answer the questions that follows, below the video.
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TABLE OF SET NOTATION (SYMBOLS) AND THE SYMBOL NAME.
SET SYMBOL SYMBOL NAME
| { } | set |
| A ∪ B | A union B |
| A ∩ B | A intersection B |
| A ⊆ B | A is subset of B |
| A ⊄ B | A is not subset B |
| A ⊂ B | proper subset / strict subset |
| A ⊃ B | proper superset / strict superset |
| A ⊇ B | superset |
| A ⊅ B | not superset |
| Ø | empty set |
| P (C) | power set |
| A = B | Equal set |
Set Operations
The five important set operations that are widely used are:
- Union of sets
- Intersection of sets
- Complement of sets
- Difference of sets
- Cardinality of a set
Union of Sets
The union of two sets X and Y is equal to the set of elements that are present in set X, in set Y, or in both the sets X and Y. This operation can be represented as;
X ∪ Y = {a: a ∈ X or a ∈ Y}
Let us consider an example, say; set A = {1, 3, 5} and set B = {1, 2, 4} then;
A ∪ B = {1, 2, 3, 4, 5}
Cardinality of a set
The number of distinct elements or members in a finite set is known as the cardinality of a set. Basically, through cardinality, we define the size of a set. The cardinality of a set A is denoted as n(A), where A is any set and n(A) is the number of members in set A.
Consider a set A consisting of the prime numbers less than 10.
Set A ={2, 3, 5, 7}.
As the set A consists of 4 elements, therefore, the cardinality of set A is given as n(A) = 4.
Intersection of Sets
The intersection of sets for two given sets is the set that contains all the elements that are common to both sets.
The intersection of sets A and B (denoted as A∩B) is the set containing the elements that are in both A and B
To represent the relationship between the sets
X = {1, 2, 5, 6, 7, 9, 10} and Y = {1, 3, 4, 5, 6, 8, 10}
Solution:
We find that X ∩ Y = {1, 5, 6, 10} ← Elements in both X and Y
Complement of Sets
The complement of A, written A’, contains all events in the universal set which are not members of set A.
For example, Set U = {2,4,6,8,10,12} and set A = {4,6,8}, then the complement of set A, A′ = {2,10,12}.
Difference of Sets
The difference of the two sets A and B is the set of elements which are present in A but not in B. It is denoted as A-B.
Example of Difference of sets
Let A = {3 , 4 , 8 , 9 , 11 , 12 } and B = {1 , 2 , 3 , 4 , 5 }. Find A – B and B – A.
Solution: We can say that A – B = { 8, 9, 11, 12} as these elements belong to A but not to B
B – A ={1,2,5} as these elements belong to B but not to A.
SET THEORY: WAEC GCE PRACTICE QUESTIONS
NOTE: TYPE YOUR ANSWERS INTO THE COMMENT BOX BELOW THESE QUESTIONS, THE CORRECTIONS WILL BE POSTED SOON.
1. If p = {1, 3, 5, 7, 9} and Q = {2, 4, 6, 8, 10} are subsets of a universal set. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. What are the elements of P1 ∩∩ Q1?
- A. {1, 3, 5, 7, 9}
- B. [2, 4, 6, 8, 10}
- C. ∅∅
- D. {1, 2, 6, 8, 10}
2. Given the sets A = [2, 4, 6, 8] and B = [2, 3, 5, 9]. If a number is selected at random from set B, what is the probability that the number is prime?
- A. 1
- B. 3/4
- C. 1/2
- D. 1/4
3. Which of the following is true for the set P={−3.2≤x<5}P={−3.2≤x<5} where x is an integer
- A. least value of x is -3.2
- B. least value of x is -3
- C. greatest value of x is 4.9
- D. greatest value of x is 5
4. M and N are two subsets of the universal set (U). If n(U) = 48, n(M) = 20, n(N) = 30 and n(MUN) = 40, find n(M ∩∩ N)
- A. 18
- B. 20
- C. 30
- D. 38
5. If p = {prime factors of 210} and Q = {prime less than 10}, find p ∩∩ Q
- A. {1,2, 3}
- B. {2, 3, 5}
- C. {1, 3, 5,7}
- D. {2,3,5,7}
6. If P is a set of all prime factors of 30 and Q is a set of all factors of 18 less than 10, find P ∩∩ Q
- A. {3}
- B. {2,3}
- C. {2,3,5}
- D. {1,2}
7. In a class of 46 students, 22 play football and 26 play volleyball. If 3 students play both games, how many play neither?
- A. 1
- B. 2
- C. 3
- D. 4
8. In a school of 150 students, 80 offer French while 60 offer Arabic and 20 offer neither. How many students offer both subjects?
- A. 45
- B. 10
- C. 35
- D. 30
9. Given U = {x: x is a positive integer less than 15} and P = {x: x is even number from 1 to 14}. Find the compliment of P.
- A. {1, 3, 5, 7, 9, 11, 13, 15}
- B. {2, 3, 5, 7, 9, 11, 13}
- C. {1, 3, 5, 7, 9, 11, 13}
- D. {2, 3, 5, 7, 11, 15}
10. Given
P = {1, 3, 5, 7, 9, 11}
and Q = {2, 4, 6, 8, 10, 12}. Determine the relationship between P and Q
- A. P∩Q = ∅
- B. P ⊂ Q
- C. Q⊂P
- D. P = Q
11. If X = {all the perfect squares less than 40}
Y = {all the odd numbers from 1 to 15}. Find X ∩ Y.
- A. {3, 9}
- B. {9}
- C. {9, 25}
- D. {1, 9}
TYPE YOUR ANSWERS INTO THE COMMENT BOX BELOW THESE QUESTIONS, THE CORRECTIONS WILL BE POSTED SOON.
