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Union of two sets

The union of two sets A and B is the set of all those elements which are either in A or in B or in both. This set is denoted by A B (read as “A union B”).
 
Symbolically, A B = {x : x A or x B}.
 
The union of two sets can be represented by a Venn diagram as shown in Figure.
 
The shaded region represents A B.
 
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Intersection of two sets

The intersection of two sets A and B is the set of all the elements which are common in A and B. This set is denoted by A B (read as “A intersection B”).
 
Symbolically, A B = {x : x A and x B}.
 
The intersection of two sets can be represented by a Venn diagram as shown in Figure. The shaded region represents A B.
 
86812.png

Some important results: Algebra of two sets

  1. If A B, then A B = A
  2. x A B x A or x B
  3. Associative law: (A B) C = A (B C)
  4. Distributive law:
    1. A (B C) = (A B) (A C);
    2. A (B C) = (A B) (A C)
  5. i. (A B) A = A and (A B) B = B;
     
    ii. (A B) A = A and (A B) B = B

Difference of two sets

The difference of two sets A and B (also called “relative complement” of B in A) is the set of all those elements of A which are not elements of B. It is denoted by A - B.
 
Symbolically, A - B = {x : x A and x B}.
 
A - B can be represented by Venn diagram as shown in Figure. The shaded region represents A - B.
 
86818.png
 
Remark: Clearly A - BB - A (as evident from above example). Hence difference of two sets is not commutative.

Symmetric difference of two sets

The symmetric difference of two sets A and B, denoted by A Δ B, is defined as A Δ B = (AB) (BA).




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