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**Set**A set is a collection or a group of well – defined objects or the items having some common characteristics.

Well defined means – Meaning should be clear whether a particular object belong to the group or not.

**Set Operations by using Venn-diagram**

There are four types of set operations:

- Union of Sets
- Intersection of Sets
- Difference of Sets
- Complement of Sets

**Union of Sets**The union of two sets A and B (A \(\cup\) B) is the set which contains all the members that belong to either A or B or both.

Mathematically, (A \(\cup\) B) = {x:x \(\in\) A or x \(\in\) B}

A \(\cup\) B (overlapping)

A \(\cup\) B (disjoint)

A \(\cup\) B \(\cup\) C (disjoint)

A \(\cup\) B \(\cup\) C (overlapping)

A \(\cup\) B \(\cup\) C (A and B overlapping, B and C overlapping)

Note: Common elements are written only once.

**Intersection of Sets**

The intersection of the two sets A and B (A \(\cap\) B) is the set of number formed by taking only common elements of both sets.

Mathematically,

(A \(\cap\) B) = {x: x \(\in\) A and x \(\in\) B}

A \(\cap\) B

A \(\cap\) B = B \(\subset\) A

A \(\cap\) B \(\cap\) C

Note: Intersection can be possible only in overlapping sets.

**Difference of Sets**

The difference of two sets A and B is the set of all the elements present only in set A. It is denoted by A – B.

B – A means difference of B and A.

Mathematically,

A – B = {x:x \(\in\) A but x \(\notin\) B}

B – A = {x:x \(\in\) B but x \(\notin\) A}

or, A – B = n_{o}(A) \(\Rightarrow\) A – A(A\(\cap\) B)

or, B – A = n_{o}(B) \(\Rightarrow\) B – A(A\(\cap\) B)

A – B

B – A

A – B

B – A

A – (B \(\cup\) C)

**Complement of Sets**

If A be the subset of a universal set U, then complement of set A (i.e. \(\bar{A}\)) is the set of all the members of universal set which do not belong to A.

\(\therefore\) \(\bar{A}\) = U – A

\(\bar{A}\) = {x:x \(\in\) U but x \(\notin\) A}

**Cardinality relation of sets**

n(A \(\cup\) B) = n(A) + n(B) \(\rightarrow\) In case of disjoint sets

n(A \(\cup\) B) = n(A) + n(B) – n(A \(\cap\) B) \(\rightarrow\) In case of overlapping sets

or, n(A \(\cup\) B) = n_{o}(A) + n_{o}(B) + n(A \(\cap\) B)

n(A\(\cap\) B) = n(A) + n(B) – n(A\(\cup\) B)

only A = n_{o}(A) = n(A) – n(A \(\cap\) B)

only B = n_{o}(B) = n(B) – n(A \(\cap\) B)

n\(\overline{A \cup B}\) = n(U) - n(A \(\cup\) B)

**Some examples**

**A, B and C are the subsets of the universal set U. If U = {1, 2, 3, …….., 15}, A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8, 10} and C = {3, 6, 9, 12, 15}. Find:(i) A \(\cup\) B \(\cup\) C(ii) A \(\cap\) B \(\cap\) C(iii) \(\overline{A \cup B \cup C}\)(iv) \(\overline{A \cap B \cap C}\)**

Solution:

U = {1, 2, 3, …….., 15}

A = {1, 2, 3, 4, 5, 6}

B = {2, 4, 6, 8, 10}

C = {3, 6, 9, 12, 15}

- A \(\cup\) B \(\cup\) C = {1, 2, 3, 4, 5, 6} \(\cup\) {2, 4, 6, 8, 10} \(\cup\) {3, 6, 9, 12, 15}

\(\therefore\) A \(\cup\) B \(\cup\) C = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15} - A \(\cap\) B \(\cap\) C = {1, 2, 3, 4, 5, 6} \(\cap\) {2, 4, 6, 8, 10} \(\cap\) {3, 6, 9, 12, 15}

\(\therefore\) A \(\cap\) B \(\cap\) C = {6} - \(\overline{A \cup B \cup C}\) = U – (A \(\cup\) B \(\cup\) C)

or, \(\overline{A \cup B \cup C}\) = {1, 2, 3, …….., 15} - {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15}

\(\therefore\) \(\overline{A \cup B \cup C}\) = {7, 11, 13, 14} - \(\overline{A \cap B \cap C}\) = U – { A \(\cap\) B \(\cap\) C}

or, \(\overline{A \cap B \cap C}\) = {1, 2, 3, …….., 15} – {6}

\(\therefore\) \(\overline{A \cap B \cap C}\) = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15}

**If n(U) = 100, n(A) = 62 a, n(B) = 42 and n(A \(\cup\) B) = 84, then find (\(\overline{A \cup B}\)), n(A\(\cap\)B), n _{o}(A) and n_{o}(B).**

Illustrate the information in venn diagram.

Solution,

Here,

n(U) = 100

n(A) = 62

n(B) = 42

n(A \(\cup\) B = 84

Now,

- (\(\overline{A \cup B}\)) = n(U) – n(A\(\cup\) B)

or, (\(\overline{A \cup B}\)) = 100 -84

\(\therefore\) (\(\overline{A \cup B}\)) = 16 - n(A\(\cap\)B) = n(A) + n(B) – n(A \(\cap\) B)

or, 84 = 62 + 42 - n(A \(\cap\) B)

or, n(A \(\cap\) B) = 104 – 84 = 20 - n
_{o}(A) = n(A) - n(A \(\cap\) B)

or, n_{o}(A) = 62 – 20

\(\therefore\) n_{o}(A) = 42 - n
_{o}(B) = n(B) - n(A \(\cap\) B)

or, n_{o}(B) = 42 - 20

\(\therefore\) n_{o}(B) = 22

**Exercise **

**(Very Short Questions = 1 marks)**

- If A = {5, 7, 9} and B = {6, 7, 8, 9, 10}, find:

(i) B –A

(ii) A \(\cap\) B - From the given venn diagram, write down the members of the sets (P – Q)\(\cup\) (Q – P).
- If U = {1, 2, 3, ………, 10}, A = {1, 3, 5, 7}, B {2, 3, 4, 8}, find:

(i) \(\overline{A \cup B}\)

(ii) \(\overline{A \cap B}\) - Prove that \(\overline{P}\) = P

**(Short Questions = 2 marks)**

- If U = {whole number less than 20}, P = {x:x is prime number}, Q = {y:y is factor of 18}, R = {z:z is multiple of 3 less than 20}, find:

(i) \(\bar{P}\) \(\cap\) \(\bar{Q}\)

(ii) \(\overline{P – Q}\)

(iii) \(\overline{P}\) \(\cup\) Q \(\cup\) R

(iv) P \(\cap\) (Q \(\cup\) R)

(v) P \(\cup\) (Q \(\cap\) R)

(vi) P \(\cup\) (Q – R) - Find the value of x from the following venn diagram.

**(Long Questions)**

- A and B ae two subsets of universal set 'U' in which n(U) = 43, n(A) = 25, n(B0 = 18 and n(A \(\cap\) B) = 7.

(i) Draw a venn diagram of above information

(ii) Find the values of n(\(\overline{A \cup B}\)) - A and B are two subsets of a universal set 'U' in which n(U) = 70, n(A) = 40, n(B) = 20 and n(\(\overline{A \cup B}\)) = 15, then

(i) show the above information in venn diagram

(ii) fin A \(\cap\) B - If n(A) = 45, n(B) = 65 and n(A \(\cup\) B) = 85, then:

(i) Find the value of n(A \(\cap\) B).

(ii) Find the value of n_{o}(B).

(iii) Represent the above information in venn diagram.

**Exercises**

- In an examination, 60% examinees failed in Mathematics, 55% failed in English and 24 failed in both subjects. If none of the examinees passed in both subjects, find:

(i) the number of examinees who passed in mathematics only.

(ii) Represent the information in venn diagram. - 60 students in a class like mathematics or science or both. Out of them 15 like both subjects. The ratio of the number of students who like mathematics to those who like science is 2:3. Find:

(i) The number of students who like mathematics

(ii) The number of students who like science only.

(iii) Represent above information in venn diagram. - In a survey of 500 people it was found that 60% of them can speak Nepali language 50% can speak Maithili language and some of the people can speak both the languages.

(i) Illustrate the above information in a venn diagram.

(ii) Find the number of people who can speak Nepali only.

(iii) Find the number of people who can speak Maithili.

(iv) Find the number of people who can speak both the languages. - Out of 100 students in an examination, 70 passed in Maths, 60 passed in Science and 20 failed in both subjects. Find the number who passed in both by using venn diagram.
- In a survey of 150 people, it was found that 80 people read daily Newspaper 120 read weekly newspaper, 60 read both types of paper and rest do not read any paper.

(i) Present the above information in venn diagram.

(ii) How many people don't read any types of the paper?

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