# Set operations: Union, Intersection, Complement and number of elements in a set.

**WEEK FIVE **

**SS1 FURTHER MATHS FIRST TERM**

**SETS**

**CONTENT**:

- Set operations: Union, Intersection, Complement and number of elements in a set.
- Venn diagram and Applications up to 3 Set Problem

**SUB TOPIC: SET OPERATONS**

**UNION OF SETS: **The union of set and is the set which consists of elements that are either in or or both. The set notation for the operation of union Thus, union is written as In set theoretical notation:

**Example: **

Given that

**INTERSECTION OF SETS:** The intersection of two sets is the set which consists of the elements that are in as well as in The set notation is written as Thus, means In set theoretical notation, the set

**Example:**

Given that

**COMPLEMENT OF A SET IN A UNIVERSAL SET: **Let be a universal set and is a subset. The complement of is the universal set written ** ^{c}** or is the set of elements in which are not contained in .

**’**

[mediator_tech]

’^{c} is the blue shaded part)

**CARDINALITY OF A SET OR NUMBER OF ELEMENTS IN A SET: **

Consider the set There are five elements in the set. The numbers of elements in a set is called cardinality. Since the set has five elements, its cardinal number is 5. The cardinality of the set is denoted Hence in the given set above, n(A)=5.

**CLASS ACTIVITIES:**

Find: (i) (ii). (iii). (iv). (v).

Find : (i). H (ii). (iii). (iv). (v). (vi).

**SUB TOPIC: VENN DIAGRAMS AND APPLICATIONS UP TO THREE SET PROBLEM**

Sets can be represented diagrammatically by closed figures. This method of set representation was developed by John Venn. A Venn diagram is therefore a pictorial representation of sets. The operations of intersection, union, complement of sets can easily be demonstrated by using Venn diagrams.

**A**

**B**

- or A is the shaded portion.

**A**

**B**

- is a subset of or

**A**

**B**

- Complement or is the shaded part.

**A**

**A’**

**A**

**B**

**A**

**B**

**A**

**B**

- is the shaded part shown below.

**A**

**B**

**APPLICATION OF VENN DIAGRAMS INVOLVING TWO SETS.**

**Examples:**

- In an examination, 18 candidates passed Mathematics, 17 candidates passed Physics, 11 candidates passed both subjects and 1 candidate failed both subjects, find:
- The number of candidates that passed mathematics only;
- The number of candidates that passed Physics only;
- The total number of candidates that sat for the examination.

Solution:

Let

Then,

Let

Then,

The solution can be represent in Venn diagram thus:

**M**

**P**

**6**

**7**

**11**

**1**

- A survey carried out on 25 adult showed that 18 of them ate fried rice, while 20 of them ate jollof rice. Find the number that ate fried rice and those that ate jollof rice, if each of them did eat least one of the two food.

Solution:

Let .

Let,

Then,

**Class Activity:**

- Represent the solution of example 2 in a Venn diagram.
- In a music competition, each competitor could play at least one brand of musical instrument. Two brands of musical instrument; keyboard and jazz band set were asked to play that day. 10 of the competitors could play key board, while 14 of the competitors could play jazz band set. If 4 of the competitors could play both brands of musical instruments, find: (i). the number of competitors that could play keyboard; (ii). the number of competitors that could play keyboard; (iii). The total number of people involved in the competition.

**Example: **

In a certain class, 22 pupils take one or more of Chemistry, Economics and Government. 12 take Economics (E), 8 take Government(G), and 7 take Chemistry(C) nobody takes Economics and Chemistry and 4 pupils take Economics and Government. (a) Using set notation and the letters indicated above, write down the statement in the last sentence. Ii. Draw a Venn diagram to illustrate the information. (b). How many pupils take: i. both Chemistry and Government. Ii. Government only.

**Solution:**

- Let

Hence, the number of pupils that took both chemistry and Government is 1. The number of pupils that took government only is

**CLASS ACTIVITIES:**

- All the 50 science students in a college in Ibadan were asked their subject combinations. 18 of the students offered further Mathematics, 21 offered Chemistry while 16 offered Biology. 7 students offered Further Mathematics and Chemistry, 8 students offered Further Mathematics and Biology, 9 students offered chemistry and biology while 5 students offered the three subject combinations. Using Venn diagrams, find: (a). the number of students that offered Further Mathematics but offered neither Chemistry nor Biology; (b). the number of students that offered Biology but offered neither Further Mathematics nor Chemistry; (c). the number of students who did not offer any of the three subject combinations.
- A survey conducted recently showed that of the 100 final year science students of International school, Accra-Ibadan, 63 entered for Agricultural science, 47 entered for Biology and 40 entered for Chemistry, 25 entered for Agricultural science and Biology, 18 entered for Agricultural science and Chemistry, while 22 entered for Biology and Chemistry. If 10 students entered for the three subject combinations, use Venn diagram to find: (i). the number of students that entered for agricultural science only; (ii). the number of students that entered for Biology only; (iii). the number of students that entered for Chemistry only; (iv). The number of students who did not enter for any of the three subject combinations.#

**PRACTICE EXERCISE:**

**Objective Test:**

- Which of the following is equivalent to
- b. c. Q d. e.
- …………….diagram is a pictorial representation of sets.
- Union b. intersection c. null d. Venn

** Essay Questions:**

- of the girls in aschool play Handball, play Volleyball. Every girl plays at least one of these games. If 27 girls play both games, how many girls are in the school?
- If
_{xx} - M
_{2}F_{6}(b). M_{2}F_{8}(c). M_{3}F_{6}(d). M_{3}F_{12} - If

**ASSIGNMENT:**

- Explain the following terms:
- Union sets. (ii) Intersection of sets (iii) Complement of set (iv) Number of a set (v) Venn diagram.
- P and Q are non-empty sets such that n(P) =10 and n(Q) = 6. Find the smallest possible value of n(PUQ).
- In a class of 40 students, 25 speak Hausa, 16 speak Igbo, 21 speak Yoruba and each of the students speaks at least one of these three languages. If 8 speak Hausa and Igbo, 11 speak Hausa and Yoruba and 6 speak Igbo and Yoruba: (a) draw a Venn a diagram to illustrate this information, using x to represent the numbers of students who speak all three; (b) calculate the value of x.
- In an examination, 31 candidates passed chemistry, 29 passed physics and 3 failed both subjects. If 50 candidates sat for the examination, how many of them passed chemistry only?
- List any three algebra laws of set and explain with the use of Venn diagram

**KEY WORDS**

**SETS****COLLECTION****ELEMENT****MEMBER****UNION****INTERSECTION****UNIVERSAL****PRIME/COMPLEMENT****VENN DIAGRAM****SUBSET****NULL SET****POWER SET****CARDINAL SET****NOTATION****OPERATIONS**