Operations on Sets

Now, we introduce some operations on sets to construct new sets from the given ones

(i) Union of Two Sets

Let A and B be two sets, then union of A B and is a set of all those elements which are in A or in Bor in both A and B . It is denoted by A $\cup$ B and read as ‘A union B’. 

Symbolically, A $\cup$ B = {x: x $\in$ A or x:x $\in$ B}

Clearly, A $\in$ A $\cup$ B
             x $\in$ A or x $\in$ B
             A $\notin$ A $\cup$ B
            x $\notin$ A or x $\notin$ B

The venn diagram of A $\cup$ B is as shown in the figure and the shaded portion represents A $\cup$ B.

e.g., If A = {1, 2, 3, 4} and B = {4, 8, 5, 6} 

So, A $\cap$ B = {4}.

General Form

The union of a finite number of sets $A_1$, $A_2$, $A_3$, . . . , $A_n$ is represented by 

$A_1 \cup  A_2 \cup A_3 \cup . . . \cup A_n$ or $\bigcup_{i = 1}^{n}A_i$

Symbolically, $\bigcup_{i = 1}^{n}A_i$ = {x: x $\in$ $A_i$ for atleast one i}

(ii) Intersection of Two Sets 

If A B and are two sets, then intersection of A B and is a set of all those elements which are in both A B and . The intersection of A B and is denoted by A $\cap$ B and read as “A intersection B”.

Symbolically, A $\cap$ B = {x: x $\in$ A or x:x $\in$ B}

If, A $\in$ A $\cap$ B ⇒ x $\in$ A or x $\in$ B

and if, A $\notin$ A $\cap$ B ⇒ x $\notin$ A or x $\notin$ B

The Venn diagram of A $\cap$ B is as shown in the figure and the shaded region represents A $\cap$ B.

e.g., If A = {1, 2, 3, 4} and B = {4, 8, 5, 6} 

So, A $\cup$ B = {1, 2, 3, 4, 5, 6, 8}.

General Form

The union of a finite number of sets $A_1$, $A_2$, $A_3$, . . . , $A_n$ is represented by 

$A_1 \cap  A_2 \cap A_3 \cap . . . \cap A_n$ or $\bigcap_{i = 1}^{n}A_i$

Symbolically, $\bigcap_{i = 1}^{n}A_i$ = {x: x $\in$ $A_i$ for all i} 

(iii) Disjoint of Two Sets

Two sets A B and are known as disjoint sets, if A $\cap$ B =  $\phi$ i.e., if A and B have no common element. The Venn diagram of disjoint sets as shown in the figure.

e.g., If A = {1, 2, 3} and B = {4, 5, 6}, then A $\cap$ B = { } = $\phi$ 

So, A and B are disjoint sets.

(iv) Difference of Two Sets 

If A and B are two non-empty sets, then difference of A and B is a set of all those elements which are in A but not in B. It is denoted as A - B. If difference of two sets is B - A, then it is a set of those elements which are in B but not in A. 

Hence, A - B = {x : x $\in$ A and x $\notin$ B} 
and B - A = {x : x $\in$ B and x $\notin$ A}  
if x $\in$ A - B ⇒ x $\in$ A but x $\notin B$
and if x $\in$ B - A ⇒ x $\in$ B but x $\notin A$

The Venn diagram of A - B and B - A are as shown in the figure and shaded region represents A - B and B - A.

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

So, A - B = {1, 2, 3} and B - A = {5, 6, 7, 8} 

Note 

• A - B $\neq$ B - A 
• A - B $\subseteq$ A and B - A $\subseteq$ B  
• A - $\phi$ = A - A = $\phi$ 
• The sets A - B and B - A are disjoint sets.

(v) Symmetric Difference of Two Sets

If A B and are two sets, then set (A - B) $\cup$ (B - A) is known as symmetric difference of sets A B and and is denoted by A BD . The Venn diagram of A BD is as shown in the figure and shaded region represents A $\Delta$ B.

e.g., A = {1, 2, 3} and B = {3, 4, 5, 6}, 

then A $\Delta$ B = (A - B) $\cup$ (B - A) 

                              = {1, 2} $\cup$ {4, 5, 6} = {1, 2, 4, 5, 6} 

Note 

•  Symmetric difference can also be written as A $\Delta$ B = (A $\cup$ B) - (A $\cap$ B) 
• A $\Delta$ B = B $\Delta$ A = (commutative)

(vi) Complement of a Set 

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Contents:

• Sets: Definition & Examples 
• Representation of Sets 
• Power Set 
• Venn Diagram 
• Complement Of a Set
• Operations on Sets 
• Laws of Algebra of Sets 
• Cardinal Number of a Finite and Infinite Set