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Let P and Q be two sets.

The union of sets P and Q is a set that consists of the elements from both the sets, P and Q.

Set P union Q consists of all the elements of sets P and Q.

Symbolically, the union of sets A and B is represented as

The Venn diagram representation of two sets A and B is as shown.

The region shaded in green represents the union of the sets, A and B. Both the circles are shaded since set A union B consists of all the elements of sets A and B.

Properties exhibited by the union of two sets:

(i)

(ii) (

(iii)

(iv)

(v)

Consider the sets P and Q,

In sets P and Q, the common element is five.

Therefore, the intersection of sets P and Q is 5.

Symbolically, the intersection of sets A and B is represented as

The Venn diagram representation of intersection of two sets A and B is as shown.

The region shaded in green represents the common elements belonging to the two sets.

Consider two sets C and D

There is no common element among the sets, C and D.

Therefore, C intersection D is a null set.

The Venn diagram representation of sets C and D is as shown.

1.

2.

3.

4.

5.

Distributive law can be verified with the help of Venn diagrams:

LHS of the equation

RHS of the equation:

Consider two sets X and Y.

Difference of sets X and Y taken in the same order (

Difference of sets Y and X taken in the same order (

It can be observed that,

Representation of set X - Y through a Venn diagram:

Only the region of set X, which is not shared by set Y, is shaded.

Representation of set Y - X through a Venn diagram:

Only the region of set Y that is not shared by set X is shaded.

The difference of any two sets A and B taken in order is represented symbolically as:

A â€“ B = {x: x âˆˆ A and x âˆ‰ B}

Similarly, B minus A is written symbolically as:

B â€“ A = {x: x âˆˆ B and x âˆ‰ A}

The complement of a set A is symbolically represented as:

The Venn diagram representation of the complement of a set is as shown.

The region shaded in green represents the complement of set A.

The complement of a set depends on the universal set.

N = {1,2,3,....}

Z = {.., -2, -1, 0 , 1, 2,....}

N' = {.., -2, -1, 0}

If A is a subset of the universal set U, then its complement Aâ€² is also a subset of U.

The complement of the union of two sets is the intersection of their complements, and the complement of the intersection of two sets is the union of their complements.

and

Law of double complementation:

Laws of empty set and universal set: