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 P^{1} ∩∩ Q^{1}?

**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.**