Complement of Set

Complement of Set

Posted by Sarkari-Form.com

Saturday, June 09, 2018

Set Theory

Friends, iss post mein aaj hum seekhenge ki kisi bhi diye gaye set ka 'Complement' kya hota hai aur kisi bhi diye gaye set ke Complement ko kaise find karte hain? So, friends...

Let's Begin...

Complement of a Set 

Kisi set ka Complement kya hota hai? -  isko samajhne ke liye chaliye ek example lete hain. Maan lijiye U ek universal set hai aur U = {1, 2, 3, 4, 5, ...}. Aur maan lijiye ek aur set hai.
E = {2, 4, 6, 8, 10, ... }. Tab E ka complement wo set hoga jisme U ke wo elements aayenge jo E mein nahi ho. Matlab,
Complement of E = U - E
                             = {1, 2, 3, 4, 5, ... } - {2, 4, 6, 8, 10, ... }
                             = {1, 3, 5, 7, 9, ... }

Kisi set ke complement ko symbolically uske naam(name) ke upar ( ' ) ya ( c ) likhkar show kiya jaata hai. So, abhi jo hamne example liya usme,
E' (or) E^c = {1, 3, 5, 7, 9, ... }.

[Note: 

1. Set A' (or) A^c ko (yaha par A koi sa bhi set ho sakta hai) set-builder form mein aise show kiya jaata hai, A' (or) A^c = {x:x ∈ U and x ∉ A}.
2. A' (or) A^c, U - A ke equal hota hai. U - A ko U\A se bhi show kiya jaata hai. Matlab
A' (or) A^c = U - A (or) U\A].

Chaliye 'Complement of Set' ko aur sahi tarike se samajhne ke liye kuch examples dekh lete hain.

Ex.1. If U = {x:x is a natural number} and A = {x:x is an odd number}. Find A'. 

Solution: U = {x:x is a natural number}
                    = {1, 2, 3, 4, 5, ... }
                A = {x:x is an odd number}
                    = {1, 3, 5, 7, 9, ... }

                A' = U - A
                    = {1, 2, 3, 4, 5, ... } - {1, 3, 5, 7, 9, ... }
                    = {2, 4, 6, 8, 10, ...}
                    = {x:x is an even number}.

Ex.2. If U = {x:x is a letter of English alphabet} and V = {x:x is a vowel of English alphabet} then find V'.

Solution: U = {x:x is a letter of English alphabet}
                    = {a, b, c, ... , x, y, z}
                V = {x:x is a vowel of English alphabet}
                    = {a, e, i, o, u}

                V' = U - V
                    = {a, b, c, ... , x, y, z} - {a, e, i, o, u}
                    = {b, c, d, ... , x, y, z}
                    = {x:x is a consonant of English alphabet}.


Ex.3. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5} and B = {1, 2, 9, 10} then find (A⋃B)' and (A⋂B)'. 

Solution: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
                A = {1, 2, 3, 4, 5}
                B = {1, 2, 9, 10}

          A⋃B = {1, 2, 3, 4, 5}⋃{1, 2, 9, 10}
                    = {1, 2, 3, 4, 5, 9, 10}
       (A⋃B)' = U - (A⋃B)
                    = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {1, 2, 3, 4, 5, 9, 10}
                    = {6, 7, 8}

          A⋂B = {1, 2, 3, 4, 5}⋂{1, 2, 9, 10}
                    = {1, 2}
       (A⋂B)' = U - (A⋂B)
                    = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {1, 2}
                    = {3, 4, 5, 6, 7, 8, 9, 10}

Venn Diagram of Complement of Set

(i) A'

Coloured part is A'.

(ii) (A⋃B)'

Coloured part is (A⋃B)'.
(If A⋂B ≠ Φ)

Coloured part is (A⋃B)'.
(If A⋂B = Φ)

(ii) (A⋂B)'

                                     

Coloured part is (A⋂B)'.
(If A⋂B ≠ Φ)

Coloured part is (A⋂B)'.
(If A⋂B = Φ)

Chaliye ab kuch complement of set se related properties/laws/rules dekh lete hain.

Some Properties of the Operation of Complement

(i) A⋃A' = U (Complement law)
(ii) A⋂A' = Φ (Complement law)
(iii) (A⋃B)' = A'⋂B' (De Morgan's law)
(iv) (A⋂B)' = A'⋃B' (De Morgan's law)
(v) (A')' = A (Double complementation law)
(vi) Φ' = U (Law of empty set)
(vii) U' = Φ (Law of universal set)


I hope ki mera ye article aap ko pasand aaya hoga. Agar aap ko mera ye article pasand aaya ho to comment karke hame bata sakte hain. Aapke dwaara kiye gaye comment se hame iss tarah ke post/article ko likne ke liye motivation milta hai.

Read Also:

• What is Set ? 
• Types of Set 
• Venn Diagrams
• Intersection of Sets
• Union of Sets
• Difference of Sets
• Symmetric Difference of Sets

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