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**Venn Diagrams**

A Venn diagram is a diagrammatic representation of ALL the possible relationships between different sets of a finite number of elements. Venn diagrams were conceived around 1880 by John Venn, an English logician, and philosopher. They are extensively used to teach Set Theory. A Venn diagram is also known as a **Primary diagram, Set diagram or Logic diagram.**

**Representation of Sets in a Venn Diagram**

It is done as per the following:

- Each individual set is represented mostly by a circle and enclosed within a quadrilateral (the quadrilateral represents the finiteness of the Venn diagram as well as the Universal set.)
- Labelling is done for each set with the setâ€™s name to indicate difference and the respective constituting elements of each set are written within the circles.
- Sets having no element in common are represented separately while those having some of the elements common within them are shown with overlapping.
- The elements are written within the circle representing the set containing them and the common elements are written in the parts of circles that are overlapped.

**Operations on Venn Diagrams**

Just like the mathematical operations on sets like Union, Difference, Intersection, Complement, etc. we have operations on Venn diagrams that are given as follows:

**Union of Sets**

Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12}. Represent A U B through a well-labeled Venn diagram.

The orange colored patch represents the common elements {6, 8} and the quadrilateral represents A U B.

**Properties of A U B**

- The commutative law holds true as A U B = B U A
- The associative law also holds true as (A U B) U C = A U (B U C)
- A U Ï† = A (Law of identity element)
- Idempotent Law â€“ A U A = A
- Law of the Universal Set U â€“ A U U = U

**The Intersection of Sets**

An intersection is nothing but the collection of all the elements that are common to all the sets under consideration. Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12} then A âˆ© B is represented through a Venn diagram as per following:

The orange colored patch represents the common elements {6, 8} as well as the A âˆ© B. The intersection of 2 or more sets is the overlapped part(s) of the individual circles with the elements written in the overlapped parts. Example:

**Properties of A âˆ© B**

- Commutative law â€“ A âˆ© B = Bâˆ© A
- Associative law â€“ (A âˆ© B)âˆ© C = A âˆ© (Bâˆ© C)
- Ï† âˆ© A = Ï†
- U âˆ© A = A
- Aâˆ© A = A; Idempotent law.
- Distributive law â€“ A âˆ© (Bâˆ© C) = (A âˆ© B) U(A âˆ© C)

**Difference of Sets**

The difference of set A and B is represented as: A â€“ B = {x: x Ïµ A and x Ïµ B} {converse holds true for B â€“ A}. Let, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A â€“ B = {1, 3, 5} and B â€“ A = {8}. The sets (A â€“ B), (B â€“ A) and (A âˆ© B) are **mutually disjoint sets.**

It means that there is NO element common to any of the three sets and the intersection of any of the two or all the three sets will result in a null or void or empty set. **A â€“ B and B â€“ A **are represented through Venn diagrams as follows:

**Complement of Sets**

If U represents the Universal set and any set A is the subset of A then the complement of set A (represented as Aâ€™) will contain ALL the elements which belong to the Universal set U but NOT to set A.

**Mathematically â€“ Aâ€™ = U â€“ A**

Alternatively, the complement of a set A, Aâ€™ is the difference between the universal set U and the set A. Example: Let universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and set A = {1, 3, 5, 7, 9}, then complement of A is given as: Aâ€™ = U â€“ A = {2, 4, 6, 8, 10}

**Properties Of Complement Sets**

A U Aâ€™ = U

A âˆ© Aâ€™ = Ï†

De Morganâ€™s Law â€“ (A U B)â€™ = Aâ€™ âˆ© Bâ€™ OR (A âˆ© B)â€™ = Aâ€™ U Bâ€™

Law of double complementation : (Aâ€™)â€™ = A

Ï†â€™ = U

Uâ€™ = Ï†

**Solved Examples For You**

**Ques 1: ****Represent the Universal Set (U) = {x : x is an outcome of a diceâ€™s roll} and set A = {s : s Ïµ Even numbers} through a Venn diagram.**

**Solution: **Since, U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}. Representing this with a Venn diagram we have:

Here, A is a subset of U, represented as â€“ A âŠ‚ U or

U is the superset of A, represented as â€“ UâŠƒ A

If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8},

then represent A â€“ B and B â€“ A through Venn diagrams.

A â€“ B = {1, 2, 3}

B â€“ A = {6, 7, 8}

Representing them in Venn diagrams:

**a. A-B**

**b. B-A**

**Operations on sets :**

When two or more sets are combined together to form another set under some given conditions, then operations on sets are carried out.**The important operations on sets are. ****Union**

Let X and Y be two sets.

Now, we can define the following new set.**X u Y = {z | z âˆˆ X or z âˆˆ Y}**

(That is, z may be in X or in Y or in both X and Y)

X u Y is read as "X union Y"

Now that X u Y contains all the elements of X and all the elements of Y and the figure given below illustrates this.

It is clear that X âŠ† X u Y and also Y âŠ† X u Y**Intersection**

Let X and Y be two sets.

Now, we can define the following new set.**X n Y = {z | z âˆˆ X and z âˆˆ Y}**

(That is z must be in both X and Y)

X n Y is read as "X intersection Y"

Now that X n Y contains only those elements which belong to both X and Y and the figure given below illustrates this.

It is trivial that that X n Y âŠ† X and also X n Y âŠ† Y**Set difference**

Let X and Y be two sets.

Now, we can define the following new set.**X \ Y = {z | z âˆˆ X but z âˆ‰ Y}**

(That is z must be in X and must not be in Y)

X \ Y is read as "X difference Y"

Now that X \ Y contains only elements of X which are not in Y and the figure given below illustrates this.

Some authors use A - B for A \ B. We shall use the notation A \ B which is widely used in mathematics for set difference. **Symmetric difference**

Let X and Y be two sets.

Now, we can define the following new set.**X Î” Y = (X \ Y) u (Y \ X)**

X Î” Y is read as "X symmetric difference Y"

Now that X Î” Y contains all elements in X u Y which are not in X n Y and the figure given below illustrates this. .

**Complement**

If X âŠ† U, where U is a universal set, then U \ X is called the compliment of X with respect to U.

If underlying universal set is fixed, then we denote U \ X by X' and it is called compliment of X.

X' = U \ X The difference set set A \ B can also be viewed as the compliment of B with respect to A.

**Disjoint sets**

Two sets X and Y are said to be disjoint if they do not have any common element. That is,

X and Y are disjoint if X n Y = áµ©

It is clear that n(A u B) = n(A) + n(B), if A and B are disjoint finite set.

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