Union of Sets
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Friends, iss article mein hum seekhenge ki sets ke Union ka kya matlab hota hai aur agar hame do(two) ya do(two) se jyada sets diye gaye ho to unka Union kaise find kar sakte hain! So, friends...
Maan lijiye A aur B koi do(two) set hain. To A aur B ka union wo set hota hai jisme set A ke sabhi elements ho aur set B ke bhi sabhi elements ho. Union ka symbol '∪' hai. Hum set A aur set B ke Union ko symbolically 'A∪B' se show karte hain aur isko 'A union B' padhte hain (read karte hain). Example, A = {1, 2, 3, 4} and B = {1, 2, 5, 6} then A∪B = {1, 2, 3, 4, 5, 6}.
[Note:
(i) Sets ka Union karte time common elements ko ek baar hi likhte hain, jaisa ki upar ke example mein aapne dekha ki 1 aur 2, ye dono set A aur set B mein hain. But A∪B mein inko sirf ek-ek baar hi likha gaya hai.
(ii) Set A aur B, chaahe A aur B koi se bhi sets ho, ke Union ko set-builder form mein aise define kiya jaata hai- A∪B = {x:x ∈ A or x ∈ B}].
Chaliye 'Union' 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}
= {a, e, i, o, u}
Ex.2. If A = {1}, B = {1, 2, 5} and C = {1, 2, 3, 6, 7} then find A∪B∪C.
Solution: A = {1}
B = {1, 2, 5}
C = {1, 2, 3, 6, 7}
A∪B∪C = {1}∪{1, 2, 5}∪{1, 2, 3, 6, 7}
= {1, 2, 3, 4, 5, 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, 7, 8, 9}
= {x:x is a natural number and 1 ≤ x < 10}.
Venn Diagram for Union of Sets
Kinhi do(two) sets, maan lijiye set A aur set B, ke liye A∪B ka Venn diagram aisa hota hai-
(i) Agar set A aur set B mein kuch common element ho (i.e., A⋂B ≠ Φ):
(ii) Agar set A aur set B mein kuch common element naa ho (i.e., A⋂B = Φ):
Kinhi teen(three) sets, maan lijiye set A, set B and set C, ke liye A∪B∪C ka diagram kuch aisa dikhta hai-
(i) Agar set A, set B aur set C mein kuch common element ho (i.e., A⋂B⋂C ≠ Φ):
(ii) Agar set A, set B aur set C mein kuch common element naa ho (i.e., A⋂B⋂C = Φ):
[Agar aap sets ke 'Intersection' ke baare mein detail mein jaanna chahte hain to neeche(below) diye gaye iss blue link par click kijiye >>>
Intersection of Sets].
Chaliye ab kuch Union se related properties/laws/rules dekh lete hain.
Some Properties of the Operation of Union
(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 union '∪'].
(iv) A∪A = A (Idempotent law)
(v) U∪A = U (Law of U)
(vi) A∪(B⋂C) = (A∪B)⋂(A∪C) (Distributive law)
[Note: ∪ distributes over ⋂].
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:
Let's Begin...
Union of Sets
Maan lijiye A aur B koi do(two) set hain. To A aur B ka union wo set hota hai jisme set A ke sabhi elements ho aur set B ke bhi sabhi elements ho. Union ka symbol '∪' hai. Hum set A aur set B ke Union ko symbolically 'A∪B' se show karte hain aur isko 'A union B' padhte hain (read karte hain). Example, A = {1, 2, 3, 4} and B = {1, 2, 5, 6} then A∪B = {1, 2, 3, 4, 5, 6}.
[Note:
(i) Sets ka Union karte time common elements ko ek baar hi likhte hain, jaisa ki upar ke example mein aapne dekha ki 1 aur 2, ye dono set A aur set B mein hain. But A∪B mein inko sirf ek-ek baar hi likha gaya hai.
(ii) Set A aur B, chaahe A aur B koi se bhi sets ho, ke Union ko set-builder form mein aise define kiya jaata hai- A∪B = {x:x ∈ A or x ∈ B}].
Chaliye 'Union' 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}
= {a, e, i, o, u}
Ex.2. If A = {1}, B = {1, 2, 5} and C = {1, 2, 3, 6, 7} then find A∪B∪C.
Solution: A = {1}
B = {1, 2, 5}
C = {1, 2, 3, 6, 7}
A∪B∪C = {1}∪{1, 2, 5}∪{1, 2, 3, 6, 7}
= {1, 2, 3, 4, 5, 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, 7, 8, 9}
= {x:x is a natural number and 1 ≤ x < 10}.
Venn Diagram for Union of Sets
Kinhi do(two) sets, maan lijiye set A aur set B, ke liye A∪B ka Venn diagram aisa hota hai-
(i) Agar set A aur set B mein kuch common element ho (i.e., A⋂B ≠ Φ):
 |
| Coloured part is A∪B |
(ii) Agar set A aur set B mein kuch common element naa ho (i.e., A⋂B = Φ):
 |
| Coloured part is A∪B |
Kinhi teen(three) sets, maan lijiye set A, set B and set C, ke liye A∪B∪C ka diagram kuch aisa dikhta hai-
(i) Agar set A, set B aur set C mein kuch common element ho (i.e., A⋂B⋂C ≠ Φ):
 |
| Coloured part is A∪B∪C |
(ii) Agar set A, set B aur set C mein kuch common element naa ho (i.e., A⋂B⋂C = Φ):
 |
| Coloured part is A∪B∪C |
Intersection of Sets].
Chaliye ab kuch Union se related properties/laws/rules dekh lete hain.
Some Properties of the Operation of Union
(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 union '∪'].
(iv) A∪A = A (Idempotent law)
(v) U∪A = U (Law of U)
(vi) A∪(B⋂C) = (A∪B)⋂(A∪C) (Distributive law)
[Note: ∪ distributes over ⋂].
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:
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