COMPLEMENT OF A SET

About "Complement of a set"

Complement of a set :

Complement is one of the important operations on sets which can be used to find the difference between the universal set and the given set. 

Let us discuss this operation in detail. 

If X ⊆ 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.  

Example : 

Let U  =  { 1, 2, 3, 4, ......10 }, A  =  { 5, 6, 7, 9 }, find A'. 

Solution :

A'  =  U \ A

A'  =  { 1, 2, 3, 4, ......10 } \ { 5, 6, 7, 9 }

A'  =  { 1, 2, 3, 4, 8, 10 }

Apart from the operation complement on sets, we have some other important operations on sets.

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

Let us discuss the other important operations here:

The other important operations on sets are. 

1. Union

2. Intersection

3. Set difference

4. Symmetric difference

5. Disjoint sets

Let us discuss the above operations in detail one by one.

Union

Let X and Y be two sets.

Now, we can define the following new set.

X u Y  =  {z | z ∈ X or z ∈ 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 ⊆ X u Y and also Y ⊆ X u Y

Intersection

Let X and Y be two sets.

Now, we can define the following new set.

X n Y  =  {z | z ∈ X and z ∈ 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 ⊆ X and also X n Y ⊆ Y

Set difference

Let X and Y be two sets.

Now, we can define the following new set.

X \ Y  =  {z | z ∈ X but z ∉  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 Δ  Y  =  (X \ Y) u (Y \ X)

X Δ Y is read as "X symmetric difference Y"

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

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 = ᵩ

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

After having gone through the stuff given above, we hope that the students would have understood "Complement of a set". 

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