Symmetric Difference of Sets

Friday, June 08, 2018

Symmetric Difference of Sets

Friday, June 08, 2018

Friends, iss article mein hum seekhenge ki sets ke Symmetric Difference ka kya matlab hota hai aur agar hame do(two) sets diye gaye ho to unka Symmetric Difference kaise find kar sakte hain! So, friends...

Let's Begin...

Symmetric Difference of Sets

Maan lijiye A aur B koi do(two) set hain. To A aur B ka Symmetric Difference (in the same order) wo set hota hai jisme set A aur set B ke common elements ko chhodkar, set A aur set B baaki ke elements aate hain. Hum set A aur set B ke Symmetric Difference ko symbolically 'A⊕B' se show karte hain aur isko 'Symmetric difference of A and B' padhte hain (read karte hain). A⊕B,
(A - B)⋃(B - A) ke equal hota hai, i.e., A⊕B = (A - B)⋃(B - A). Example ke liye, maan lijiye
A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then,

A - B = {1, 2, 3, 4, 5, 6} - {2, 4, 6, 8}
= {1, 3, 5}

B - A = {2, 4, 6, 8} - {1, 2, 3, 4, 5, 6}
= {8}

So, A⊕B = (A - B)⋃(B - A)
= {1, 3, 5}⋃{8}
= {1, 3, 5, 8}

[Note:

(i) Set A aur B, chaahe A aur B koi se bhi sets ho, ke Symmetric Difference ko set-builder form mein aise define kiya jaata hai- A⊕B = {x : (x ∈ A and x ∉ B) or (x ∈ B and x ∉ A)}
(ii) A⊕B ko AΔB se bhi denote kiya jaata hai].

Chaliye 'Symmetric Difference of Sets' ko aur better tarike se samajhne ke liye kuch aur examples dekh lete hain.

Ex.1. A = {a, e, i, o, u} and B = {a, i, u}. Find A⊕B.

Solution: A = {a, e, i, o, u}
B = {a, i, u}

A - B = {a, e, i, o, u} - {a, i, u}
= {e, o}

B - A = {a, i, u} - {a, e, i, o, u}
= { }
= Φ

A⊕B = (A - B)⋃(B - A)
= {e, o}⋃Φ
= {e, o}

Ex.2. If A = {1}, B = {1, 2, 5} and C = {1, 2, 3, 6, 7} then find A⊕B, B⊕C and A⊕C.

Solution: A = {1}
B = {1, 2, 5}
C = {1, 2, 3, 6, 7}

A - B = {1} - {1, 2, 5}
= { }
= Φ
B - A = {1, 2, 5} - {1}
= {2, 5}
A⊕B = (A - B)⋃(B - A)
= Φ⋃{2, 5}
= {2, 5}

B - C = {1, 2, 5} - {1, 2, 3, 6, 7}
= {5}
C - B = {1, 2, 3, 6, 7} - {1, 2, 5}
= {3, 6, 7}
B⊕C = (B - C)⋃(C - B)
= {5}⋃{3, 6, 7}
= {5, 3, 6, 7}

C - A = {1, 2, 3, 6, 7} - {1}
= {2, 3, 6, 7}
A - C = {1} - {1, 2, 3, 6, 7}
= { }
= Φ
C⊕A = (C - A)⋃(A - C)
= {2, 3, 6, 7}⋃Φ
= {2, 3, 6, 7}

Ex.3. If set P = {x:x is a natural number and 1 < x ≤ 6} and set Q = {x:x is a natural number and 6 < x < 10}. Find P - Q.

Solution: P = {x:x is a natural number and 1 < x ≤ 6}
= {1, 2, 3, 4, 5, 6}
Q = {x:x is a natural number and 6 < x < 10}
= {7, 8, 9}

P - Q = {1, 2, 3, 4, 5, 6} - {7, 8, 9}
= {1, 2, 3, 4, 5, 6}
Q - P = {7, 8, 9} - {1, 2, 3, 4, 5, 6}
= {7, 8, 9}

P⊕Q = (P - Q)⋃(Q - P)
= {1, 2, 3, 4, 5, 6}⋃{7, 8, 9}
= {1, 2, 3, 4, 5, 6, 7, 8, 9}
= {x:x is a natural number and 1 ≤ x < 10}

Venn Diagram for Symmetric Difference of Sets

Kinhi do(two) sets, maan lijiye set A aur set B, ke liye A⊕B aur B⊕A ka Venn diagram kuch aisa hota hai-

For A⊕B (or) B⊕A:

(i) Agar set A aur set B mein kuch common element ho (i.e., A⋂B ≠ Φ):

venn diagram for symmetric difference for two sets

Coloured part is A⊕B (or) B⊕A.

(ii) Agar set A aur set B mein kuch common element naa ho (i.e., A⋂B = Φ):

Coloured part is A⊕B (or) B⊕A.

Chaliye ab kuch Symmetric Difference se related properties/laws/rules dekh lete hain.

Some Properties of the Operation of Symmetric Difference

(i) A⊕B = B⊕A (Commutative law)
(ii) (A⊕B)⊕C = A⊕(B⊕C) (Associative law)
(iii) A⊕Φ = A (Law of identity element)

[Note: Φ is the identity of symmetric difference '⊕'].

(iv) A⊕A = Φ

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.

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