What is Relation ?
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Hum apni rozmarra ki zindagi (daily life) mein bahut saare aise patterns dekhte hain jinke beech mein relation (संबन्ध) hota hai jaise ki brother and sister, mother and daughter, teacher and student etc. Inn sabhi ke beech ek relation hai. Isi tarah se Mathematics mein bhi hame bahut saare cheezein dekhne ko milti hain jinke beech mein relation hota hai. For example - 4 is less than 6, line AB is parallel to line CD, set A is subset of set B. Inn sabhi ke beech ek relation hai. Relation ko precisely and formal way mein samajhne ke liye iss article ko poora padhiye.
Let's Begin...
"A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian Product A✕B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A✕B. The first element is called the 'pre-image' and the second element is called the 'image' of the pre-image (i.e., first element)".
Matlab, ek relation R ek subset hota hai Cartesian Product A✕B ka jaha par A aur B dono hi non-empty sets hain. Aur ye subset A✕B mein ordered pairs ke first aur second element/member ke beech mein ek relationship (संबन्ध) describe karke prapt(derive) kiya jaata hai. Isme first element/member ko 'pre-image' aur second element/member ko 'image' bolte hain.
Relation ko aur sahi tarike se samajhne ke liye chaliye ek example dekh lete hain.
Maan lijiye A = {1, 2, 4, 10} aur B = {7, 8, 9} hai. Tab A✕B hoga,
A✕B = {1, 2, 4, 10} ✕ {7, 8, 9}
= {(1, 7), (1, 8), (1, 9), (2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9), (10, 7), (10, 8), (10, 9)}
Ab hame ek relation banana hai jisme first element/member kam (less) ho second element/member se, matlab R = {(x, y) : x is less than y, x ∈ A and y ∈ B}. Iss relation ko banane ke liye hame A✕B mein sabhi ordered pairs ko dekhna hoga aur unme se wo ordered pairs select karne honge jisme first element/member, second element/member se kam (less) ho aur phir unka ek set banana hoga jiska naam(name) R hoga.
So, R = {(1, 7), (1, 8), (1, 9), (2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9)}
Yahi wo relation hai jisko hame banana tha. Iss relation R ko hum picture ke jariye aur achchhe (अच्छे) se samajh sakte hain. Agar kisi set ko hum picture (चित्र) ke form mein show karein to iss picture ko 'arrow diagram' kehte hain.
Relation R = {(1, 7), (1, 8), (1, 9), (2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9)} ka arrow diagram banane ke liye sabse pehle ek circle ya oval draw kijiye aur iska naam 'A' rakh dijiye. Iss circle ke side mein ek aur circle ya oval banaaiya aur iska naam 'B' rakh dijiye. Ab circle ya oval A ke ander wo sabhi elements likh dijiye jo set A mein hain. Isi tarah se circle ya oval B ke andar wo sabhi elements likh dijiye jo set B mein hain. Ab set R ke sabhi ordered pairs ko dekhiye aur set A ke jiss element/member ka relationship set B ke jiss element/member ke saath hai usko arrow (तीर) se milaa dijiye.
Kisi bhi relation mein agar sabhi ordered pairs ke first element ka set bana liya jaaye to uss set ko uss relation ka 'domain' kehte hain aur agar uss relation ke sabhi ordered pairs ke second elements ka set bana liya jaaye to uss set ko uss relation ka 'range' kehte hain. Aur kisi relation mein ordered pairs ke second element/members jiss set ke elements hote hain uss set ko uss relation ka 'codomain' kehte hain.
Abhi jo hamne relation ka example diya matlab relation R = {(1, 7), (1, 8), (1, 9), (2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9)} mein,
Domain = {1, 1, 1, 2, 2, 2, 4, 4, 4}
= {1, 2, 4}
Range = {7, 8, 9, 7, 8, 9, 7, 8, 9}
= {7, 8, 9}
Codomain = B
= {7, 8, 9}
[Note:
1. Range ⊆ Codomain always.
2 'A relation R from A to A is also stated as a relation on A'.
Matlab A se A tak koi relation R ko 'relation on A' bhi kaha jaata hai.
3. Agar set A mein p element ho aur set B mein q elements ho to A se B tak jitne bhi relation bann sakte hain unki sankhya(number) hote hain: 2pq. That means if n(A) = p and n(B) = q then total number of relations from A to B is 2pq.
Chaliye relation se related kuch aur examples dekh lete hain:
Ex.1. Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b) : a, b ∈ A, b is exactly divisible by a}.
(i) Write R in roster form.
(ii) Find the domain of R.
(iii) Find the range of R.
Solution: A✕A = {1, 2, 3, 4, 6} ✕ {1, 2, 3, 4, 6}
= {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 6)}.
(i) R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}.
(ii) Domain = {1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6}
= {1, 2, 3, 4, 6}
(iii) Range = {1, 2, 3, 4, 6, 2, 4, 6, 3, 6, 4, 6}
= {1, 2, 3, 4, 6}
Ex.2. If P = {a, b} and Q = {Anuj, Bheem, Vishal, Ankur, Billu}. Define a relation from P to Q by R = {(x, y) : x is the first letter of the name y, x ∈ P, y ∈ Q}.Write down its domain, codomain and range. Also draw arrow diagram of the relation.
Solution: P = {a, b}
Q = {Anuj, Bheem, Vishal, Ankur, Billu}
So, R= {(a, Anuj), (a, Ankur), (b, Bheem), (b, Billu)}
Domain = {a, a, b, b}
= {a, b}
Codomain = Q
= {Anuj, Bheem, Vishal, Ankur, Billu}
Range = {Anuj, Ankur, Bheem, Billu}
[Note: Ye jaroori nahi hai ki relation find karne ke liye hame pehle Cartesian Product ko find karna hoga. Bina Cartesian Product find kiye bhi hum relation find kar sakte hain jaise ki hamne Ex.2. mein kiya].
Ex.3. Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.
Solution: R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}
= {(0, 0 + 5), (1, 1 + 5), (2, 2 + 5), (3, 3 + 5), (4, 4 + 5), (5, 5 + 5)}
= {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}
Domain = {0, 1, 2, 3, 4, 5}
Range = {5, 6, 7, 8, 9, 10}
Read Also:
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.
THANKS FOR READING THIS BLOG.
Let's Begin...
What is Relation?
"A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian Product A✕B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A✕B. The first element is called the 'pre-image' and the second element is called the 'image' of the pre-image (i.e., first element)".
Matlab, ek relation R ek subset hota hai Cartesian Product A✕B ka jaha par A aur B dono hi non-empty sets hain. Aur ye subset A✕B mein ordered pairs ke first aur second element/member ke beech mein ek relationship (संबन्ध) describe karke prapt(derive) kiya jaata hai. Isme first element/member ko 'pre-image' aur second element/member ko 'image' bolte hain.
Relation ko aur sahi tarike se samajhne ke liye chaliye ek example dekh lete hain.
Maan lijiye A = {1, 2, 4, 10} aur B = {7, 8, 9} hai. Tab A✕B hoga,
A✕B = {1, 2, 4, 10} ✕ {7, 8, 9}
= {(1, 7), (1, 8), (1, 9), (2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9), (10, 7), (10, 8), (10, 9)}
Ab hame ek relation banana hai jisme first element/member kam (less) ho second element/member se, matlab R = {(x, y) : x is less than y, x ∈ A and y ∈ B}. Iss relation ko banane ke liye hame A✕B mein sabhi ordered pairs ko dekhna hoga aur unme se wo ordered pairs select karne honge jisme first element/member, second element/member se kam (less) ho aur phir unka ek set banana hoga jiska naam(name) R hoga.
So, R = {(1, 7), (1, 8), (1, 9), (2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9)}
Yahi wo relation hai jisko hame banana tha. Iss relation R ko hum picture ke jariye aur achchhe (अच्छे) se samajh sakte hain. Agar kisi set ko hum picture (चित्र) ke form mein show karein to iss picture ko 'arrow diagram' kehte hain.
Relation R = {(1, 7), (1, 8), (1, 9), (2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9)} ka arrow diagram banane ke liye sabse pehle ek circle ya oval draw kijiye aur iska naam 'A' rakh dijiye. Iss circle ke side mein ek aur circle ya oval banaaiya aur iska naam 'B' rakh dijiye. Ab circle ya oval A ke ander wo sabhi elements likh dijiye jo set A mein hain. Isi tarah se circle ya oval B ke andar wo sabhi elements likh dijiye jo set B mein hain. Ab set R ke sabhi ordered pairs ko dekhiye aur set A ke jiss element/member ka relationship set B ke jiss element/member ke saath hai usko arrow (तीर) se milaa dijiye.
Kisi bhi relation mein agar sabhi ordered pairs ke first element ka set bana liya jaaye to uss set ko uss relation ka 'domain' kehte hain aur agar uss relation ke sabhi ordered pairs ke second elements ka set bana liya jaaye to uss set ko uss relation ka 'range' kehte hain. Aur kisi relation mein ordered pairs ke second element/members jiss set ke elements hote hain uss set ko uss relation ka 'codomain' kehte hain.
Abhi jo hamne relation ka example diya matlab relation R = {(1, 7), (1, 8), (1, 9), (2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9)} mein,
Domain = {1, 1, 1, 2, 2, 2, 4, 4, 4}
= {1, 2, 4}
Range = {7, 8, 9, 7, 8, 9, 7, 8, 9}
= {7, 8, 9}
Codomain = B
= {7, 8, 9}
[Note:
1. Range ⊆ Codomain always.
2 'A relation R from A to A is also stated as a relation on A'.
Matlab A se A tak koi relation R ko 'relation on A' bhi kaha jaata hai.
3. Agar set A mein p element ho aur set B mein q elements ho to A se B tak jitne bhi relation bann sakte hain unki sankhya(number) hote hain: 2pq. That means if n(A) = p and n(B) = q then total number of relations from A to B is 2pq.
Chaliye relation se related kuch aur examples dekh lete hain:
Ex.1. Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b) : a, b ∈ A, b is exactly divisible by a}.
(i) Write R in roster form.
(ii) Find the domain of R.
(iii) Find the range of R.
Solution: A✕A = {1, 2, 3, 4, 6} ✕ {1, 2, 3, 4, 6}
= {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 6)}.
(i) R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}.
(ii) Domain = {1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6}
= {1, 2, 3, 4, 6}
(iii) Range = {1, 2, 3, 4, 6, 2, 4, 6, 3, 6, 4, 6}
= {1, 2, 3, 4, 6}
Ex.2. If P = {a, b} and Q = {Anuj, Bheem, Vishal, Ankur, Billu}. Define a relation from P to Q by R = {(x, y) : x is the first letter of the name y, x ∈ P, y ∈ Q}.Write down its domain, codomain and range. Also draw arrow diagram of the relation.
Solution: P = {a, b}
Q = {Anuj, Bheem, Vishal, Ankur, Billu}
So, R= {(a, Anuj), (a, Ankur), (b, Bheem), (b, Billu)}
Domain = {a, a, b, b}
= {a, b}
Codomain = Q
= {Anuj, Bheem, Vishal, Ankur, Billu}
Range = {Anuj, Ankur, Bheem, Billu}
[Note: Ye jaroori nahi hai ki relation find karne ke liye hame pehle Cartesian Product ko find karna hoga. Bina Cartesian Product find kiye bhi hum relation find kar sakte hain jaise ki hamne Ex.2. mein kiya].
Ex.3. Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.
Solution: R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}
= {(0, 0 + 5), (1, 1 + 5), (2, 2 + 5), (3, 3 + 5), (4, 4 + 5), (5, 5 + 5)}
= {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}
Domain = {0, 1, 2, 3, 4, 5}
Range = {5, 6, 7, 8, 9, 10}
Read Also:
- What is Set ?
- Types of Set
- Venn Diagrams
- Intersection of Sets
- Difference of Sets
- Symmetric Difference of Sets
- What are Intervals ?
- What is Cartesian Product ?
- What is a Function?
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.
THANKS FOR READING THIS BLOG.
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