What is a Function ?
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Function - ye Mathematics mein ek bahut hi important concept hai. ye word - 'function' aapko Mathematic ki har(every) branch mein dekhne ko milega. Higher level Mathematics mein to ye word bahut hi common hai. Iss word - 'function' ka use aapko higher level Mathematics mein har(every) jagah par dekhne ko milega. To aise mein ye jaroori ho jaata hai ki hame 'function' ke baare mein knowledge ho. So friends...
Let's begin...
Function ek special type ka relation hota hai. Function ko hum ek rule ki tarah dekh sakte hain jiske dwara hum kuch diye gaye elements se naye elements bana sakte hain. Aise bahut saare terms hain jaise ki - 'map' ya 'mapping' jinka use function ko denote karne ke liye kiya jaata hai.
Definition of Function
"A relation 'f' from a set A to a set B is said to be function if every element of set A has one and only one image in set B".
Matlab, ek relation 'f', jo ek set A se kisi doosre set B ko hai, function kehlaata hai agar set A ke har(every) element ka set B mein ek aur sirf ek hi image ho.
Kisi function ko mathematically denote karne ke liye hame do(two) cheezien bataani padti hain:
(i) Wo function kis set se kis set tak hai.
(ii) Wo rule jo pehle set ke har(every) element ko doosre set ke ek unique element se relate karta hai.
Example ke liye, maan lijiye ek function hai jo set A se set B tak hai. Aur set A = {1, 2, 3, 4, 5} and B = {2, 3, 4, 5, 6, 7} aur set A ka har(every) element ka relation set B mein uss element se hai jo set A ke element se ek jyada hai. Matlab set A ka har(every) element set B ke ek unique element se rule f(x) = x + 1 se related hai. Iss example mein ye wo do(two) cheezein hain jinke through hum iss function ko mathematically show kar sakte hain:
(i) Ye function f, set A se set B tak hai. Short form mein iss statement ko aise likhte hain-
'f : A → B'.
(ii) Iss function f ka rule hai f(x) = x + 1, iss rule ke through hi set A ke sabhi elements set B ke elements se related hai. Isme x ki jagah par set A koi element rakhna hota hai phir isko solve karne par f(x) ki jo value aati hai wo set B ka koi element hota hai aur isi element se set A ka wo element related hota hai. Jaise ki rule f(x) = x + 1 mein x ki jagah par set A ka element 1 rakhne par,
f(1) = 1 + 1 = 2. Solve karne ke baad f(1) ki value 2 aayi, iska matlab set A ka element 1 set B ke element 2 se related hai. Isi tarah se x ki jagah set A ka element 2 rakhne par f(2) = 2 + 1 = 3. Iska matlab set A ka element 2, set B ke element 3 se related hai. Isi tarah se set A ke element 3, 4 aur 5 ka relation set B ke element respectively 4, 5 aur 6 se hai. Arrow diagram se isko aise show kar sakte hain:
So, upar jo hamne iss function ka example liya usko mathematically aise show kar sakte hain:
'f : A → B, f(x) = x + 1'.
Iss function mein set A ka element set B ke kisi unique element se related hai; set B ke uss unique element ka 'preimage' kehlaata hai aur set B ka wo unique element set A ka 'image' kehlaata hai. For example aap arrow diagram mein ye clearly dekh sakte hain ki set A ka element 1, set B ke element 2 se related hai. So 2, 1 ka 'preimage' hai aur iss tarah 2, 1 ka 'image' hai. Isi tarah se 2, 3 ka 'preimage' hai aur 3, 2 ka 'image' hai. 3, 4 ka 'preimage' hai aur 4, 3 ka 'image' hai. 4, 5 ka 'preimage' hai aur 5, 4 ka 'image' hai. 5, 6 ka 'preimage' hai aur 6, 5 ka 'image' hai.
Kisi function mein jiss set ka relation kisi doosre set se hota hai, uss set ko function ka 'Domain' kehte hain. Aur uss set ko jissse Domain ka relationship hota hai, uss function ka 'Codomain' kehte hain. Example ke liye abhi jo hamne function ka example liya hai usme set A = {1, 2, 3, 4, 5} domain hai aur set B = {2, 3, 4, 5, 6, 7} codomain hai.
Agar kisi function ke codomain mein se sirf unn elements ko nikaalkar jinka relationship domain ke elements se hai, unka set bana liya jaaye to uss set ko function ka 'Range' kahenge. Upar jo hamne function ka example liya hai usme codomain ke elements ko dekhiye ki kaun-kaun se elements uss function ke domain se related hain (Ye aap uss function ke arrow diagram mein dekhkar aasaani se pata laga sakte hain. To aapne dekha ki 2, 3, 4, 5 aur 6 - ye element hi domain ke kisi naa kisi element se related hain. So, uss function ki range hogi, Range = {2, 3, 4, 5, 6}.
Range ko kabhi-kabhi 'R' se bhi denote karte hain. So, R = {2, 3, 4, 5, 6}.
(Dhyaan rahe ki yaha par R ka matlab range hai naa ki relation).
[Note:
(i) Agar kisi function ka domain real number (R) ya real number (R) ka koi subset hai to uss function ko 'real function' kehte hain.
(ii) Agar kisi function ka range real number (R) ya real number (R) ka koi subset hai to uss function ko 'real valued function' kehte hain].
Chaliye abhi jo kuch bhi hamne padha usko aur deeply samjhane ke liye kuch examples dekh lete hain:
Ex.1. If function is defined by f : N ➝ N, f(x) = x + 2. Draw arrow diagram of this function. Also find its domain, codomain and range.
Solution: f : N ➝ N, f(x) = x + 2
Ab f(x) mein x ki jagah par 1 rakhne par aur solve karne par,
f(1) = 1 + 2
= 3
Matlab, domain ke element 1 ka relationship codomain ke element 3 se hai.
Isi tarah se f(x) mein x ki jagah par 2 rakhne par aur solve karne par,
f(2) = 2 + 2
= 4
Matlab, domain ke element 2 ka relationship codomain ke element 4 se hai.
Isi tarah se f(x) mein x ki jagah par 3 rakhne par aur solve karne par,
f(3) = 3 + 2
= 5
Matlab, domain ke element 3 ka relationship codomain ke element 5 se hai.
Isi tarah se f(x) mein x ki jagah par 4 rakhne par aur solve karne par,
f(4) = 4 + 2
= 6
Matlab, domain ke element 4 ka relationship codomain ke element 6 se hai.
Isi tarah se f(x) mein x ki jagah par 5 rakhne par aur solve karne par,
f(5) = 5 + 2
= 7
Matlab, domain ke element 5 ka relationship codomain ke element 7 se hai.
Isi tarah se aap domain ke sahbhi elements ka codomain mein image ya relationship nikaal sakta hain.
So, Domain = N
= {1, 2, 3, 4, 5, ...}
Codomain = N
= {1, 2, 3, 4, 5, ...}
Range = N - {1, 2}
= {3, 4, 5, 6, 7, ...}
(Yaha par N mein se 1 aur 2 ko minus kar diya kyuki domain mein koi aise elements nahi hain jo inse related ho).
Ex.2. A function is defined by g : {1, 2, 3} ➝ {-1, -2, -3, -4}, g(x) = -x. Draw its arrow diagram. Also find its domain, codomain and range.
Solution: g : {1, 2, 3} ➝ {-1, -2, -3, -4}
Ab g(x) mein x ki jagah par 1 rakhne par aur solve karne par,
g(1) = -1
Matlab, domain ke element 1 ka relationship codomain ke element -1 se hai.
Isi tarah se g(x) mein x ki jagah par 2 rakhne par aur solve karne par,
g(2) = -2
Matlab, domain ke element 2 ka relationship codomain ke element -2 se hai.
Isi tarah se g(x) mein x ki jagah par 3 rakhne par aur solve karne par,
g(3) = -3
Matlab, domain ke element 3 ka relationship codomain ke element -3 se hai.
So, Domain = {1, 2, 3}
Codomain = {-1, -2, -3, -4}
Range = {-1, -2, -3}
Ex.3. State if the relation {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)} a function? If it is a function, determine its domain and range.
Solution: Relation = {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}.
Chaliye iss relation ka arrow diagram banaate hain:
Aap ye clearly dekh sakte hai ki iss relation ke domain ke har(every) element ka relationship iss set ke codomain ke sirf ek hi element (unique element) se hai. So, ye relation ek function hai. So, iss function ka domain wahi hoga jo iss relation ka domain hai.
Domain of function = {2, 5, 8, 11, 14}
Iss function ka range bhi wahi hoga jo iss relation ka range hai.
Range of function = {1}.
Ex.3. State if the relation {(1, 3), (1, 5), (2, 5)} a function? If it is a function, determine its domain and range.
Solution: Relation = {(1, 3), (1, 5), (2, 5)}.
Chaliye iss relation ka arrow diagram banaate hain:
Aap ye clearly dekh sakte hai ki iss relation ke domain ke har(every) element ka relationship iss set ke codomain ke sirf ek hi element (unique element) se nahi hai. Jaise ki iss relation ka domain ke element 1 ka relationship iss relation ke codomain ke do(two) elements 3 aur 5 se hai, so yaha par ek unique relationship nahi bann raha hai. Agar 1 ka relationship sirf 3 se hota ya phir 1 ka relationship sirf 5 se hota to isko unique relationship kaha jaata. So, diya gaya ye relation ek function nahi hai. So, iss function ka domain wahi hoga jo iss relation ka domain hai.
Ex.5. A function is Defined by f : Z - {0}➝ N, f(x) = x2. State f(-3), f(5) and f(-10). Draw its arrow diagram. Also state its domain, codomain and range.
Solution: f : Z - {0}➝ N, f(x) = x2
Ab f(x) mein x ki jagah par -3 rakhne par aur solve karne par,
f(-3) = (-3)2
= 9
Matlab, domain ke element -3 ka relationship codomain ke element 9 se hai.
Isi tarah se f(x) mein x ki jagah par 5 rakhne par aur solve karne par,
f(5) = (5)2
= 25
Matlab, domain ke element 5 ka relationship codomain ke element 25 se hai.
Isi tarah se f(x) mein x ki jagah par -10 rakhne par aur solve karne par,
f(-10) = (-10)2
= 100
Matlab, domain ke element -10 ka relationship codomain ke element 100 se hai.
Isi tarah se aap domain ke sahbhi elements ka codomain mein image ya relationship nikaal sakta hain.
2. f(x) ya g(x) ya h(x) etc. ko sirf ek single letter jaise ki 's' ya 't' ya 'y' ya 'z' etc. se bhi show kiya jaata hai. Example ke liye, function f : R ➝ R, f(x) = 3x2 + 4x - 8 mein f(x) ko y bhi likh sakte hain [i.e., y = f(x)]. Aisa karne par ye function aisa ho jaayega, f : R ➝ R, y = 3x2 + 4x - 8.
3. Agar practically kaha jaaye to - " If any quantity depends on the other quantity, then the first quantity is called the function of the other quantity".
Matlab, agar koi cheez kisi doosre cheez par depend karti hai to ye kaha jaata hai ki pehli cheez doosri cheez ka function hai. For example, hamari age (उम्र), time par depend karti hai. Jaise-jaise time aage chalta jaata hai waise-waise hamaari age bhi change hoti jaati hai. So, Mathematics ki language mein hum ye kahenge ki ' Hamaari age, time ka function hai'. Aur isko Mathematics ki language mein aise show karte hain: 'our age = f(time).' Isi tarah se kisi circle ka area uski radius par depend karta hai. Agar kisi circle ki radius kam kar di jaaye to uska area bhi kam ho jaayega aur agar uss circle ki radius jyada kar di jaaye to uska area bhi jyada ho jaayega. So, Mathematics ki language mein hum ye kahenge ki 'Kisi circle ka area, uske radius ka function hai'. Aur isko Mathematics ki language mein aise show karte hain: 'Area of circle = f(radius).' Isi tarah se agar koi quantity 'y', kisi doosri quantity 'x' par depend karta. So, Mathematics ki language mein hum ye kahenge ki 'y, x ka function hai'. Aur isko Mathematics ki language mein aise show karte hain: 'y = f(x)'].
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:
THANKS FOR READING THIS BLOG.
Let's begin...
What is a Function?
Function ek special type ka relation hota hai. Function ko hum ek rule ki tarah dekh sakte hain jiske dwara hum kuch diye gaye elements se naye elements bana sakte hain. Aise bahut saare terms hain jaise ki - 'map' ya 'mapping' jinka use function ko denote karne ke liye kiya jaata hai.
Definition of Function
"A relation 'f' from a set A to a set B is said to be function if every element of set A has one and only one image in set B".
Matlab, ek relation 'f', jo ek set A se kisi doosre set B ko hai, function kehlaata hai agar set A ke har(every) element ka set B mein ek aur sirf ek hi image ho.
Kisi function ko mathematically denote karne ke liye hame do(two) cheezien bataani padti hain:
(i) Wo function kis set se kis set tak hai.
(ii) Wo rule jo pehle set ke har(every) element ko doosre set ke ek unique element se relate karta hai.
Example ke liye, maan lijiye ek function hai jo set A se set B tak hai. Aur set A = {1, 2, 3, 4, 5} and B = {2, 3, 4, 5, 6, 7} aur set A ka har(every) element ka relation set B mein uss element se hai jo set A ke element se ek jyada hai. Matlab set A ka har(every) element set B ke ek unique element se rule f(x) = x + 1 se related hai. Iss example mein ye wo do(two) cheezein hain jinke through hum iss function ko mathematically show kar sakte hain:
(i) Ye function f, set A se set B tak hai. Short form mein iss statement ko aise likhte hain-
'f : A → B'.
(ii) Iss function f ka rule hai f(x) = x + 1, iss rule ke through hi set A ke sabhi elements set B ke elements se related hai. Isme x ki jagah par set A koi element rakhna hota hai phir isko solve karne par f(x) ki jo value aati hai wo set B ka koi element hota hai aur isi element se set A ka wo element related hota hai. Jaise ki rule f(x) = x + 1 mein x ki jagah par set A ka element 1 rakhne par,
f(1) = 1 + 1 = 2. Solve karne ke baad f(1) ki value 2 aayi, iska matlab set A ka element 1 set B ke element 2 se related hai. Isi tarah se x ki jagah set A ka element 2 rakhne par f(2) = 2 + 1 = 3. Iska matlab set A ka element 2, set B ke element 3 se related hai. Isi tarah se set A ke element 3, 4 aur 5 ka relation set B ke element respectively 4, 5 aur 6 se hai. Arrow diagram se isko aise show kar sakte hain:
So, upar jo hamne iss function ka example liya usko mathematically aise show kar sakte hain:
'f : A → B, f(x) = x + 1'.
Iss function mein set A ka element set B ke kisi unique element se related hai; set B ke uss unique element ka 'preimage' kehlaata hai aur set B ka wo unique element set A ka 'image' kehlaata hai. For example aap arrow diagram mein ye clearly dekh sakte hain ki set A ka element 1, set B ke element 2 se related hai. So 2, 1 ka 'preimage' hai aur iss tarah 2, 1 ka 'image' hai. Isi tarah se 2, 3 ka 'preimage' hai aur 3, 2 ka 'image' hai. 3, 4 ka 'preimage' hai aur 4, 3 ka 'image' hai. 4, 5 ka 'preimage' hai aur 5, 4 ka 'image' hai. 5, 6 ka 'preimage' hai aur 6, 5 ka 'image' hai.
Kisi function mein jiss set ka relation kisi doosre set se hota hai, uss set ko function ka 'Domain' kehte hain. Aur uss set ko jissse Domain ka relationship hota hai, uss function ka 'Codomain' kehte hain. Example ke liye abhi jo hamne function ka example liya hai usme set A = {1, 2, 3, 4, 5} domain hai aur set B = {2, 3, 4, 5, 6, 7} codomain hai.
Agar kisi function ke codomain mein se sirf unn elements ko nikaalkar jinka relationship domain ke elements se hai, unka set bana liya jaaye to uss set ko function ka 'Range' kahenge. Upar jo hamne function ka example liya hai usme codomain ke elements ko dekhiye ki kaun-kaun se elements uss function ke domain se related hain (Ye aap uss function ke arrow diagram mein dekhkar aasaani se pata laga sakte hain. To aapne dekha ki 2, 3, 4, 5 aur 6 - ye element hi domain ke kisi naa kisi element se related hain. So, uss function ki range hogi, Range = {2, 3, 4, 5, 6}.
Range ko kabhi-kabhi 'R' se bhi denote karte hain. So, R = {2, 3, 4, 5, 6}.
(Dhyaan rahe ki yaha par R ka matlab range hai naa ki relation).
[Note:
(i) Agar kisi function ka domain real number (R) ya real number (R) ka koi subset hai to uss function ko 'real function' kehte hain.
(ii) Agar kisi function ka range real number (R) ya real number (R) ka koi subset hai to uss function ko 'real valued function' kehte hain].
Chaliye abhi jo kuch bhi hamne padha usko aur deeply samjhane ke liye kuch examples dekh lete hain:
Ex.1. If function is defined by f : N ➝ N, f(x) = x + 2. Draw arrow diagram of this function. Also find its domain, codomain and range.
Solution: f : N ➝ N, f(x) = x + 2
Ab f(x) mein x ki jagah par 1 rakhne par aur solve karne par,
f(1) = 1 + 2
= 3
Matlab, domain ke element 1 ka relationship codomain ke element 3 se hai.
Isi tarah se f(x) mein x ki jagah par 2 rakhne par aur solve karne par,
f(2) = 2 + 2
= 4
Matlab, domain ke element 2 ka relationship codomain ke element 4 se hai.
Isi tarah se f(x) mein x ki jagah par 3 rakhne par aur solve karne par,
f(3) = 3 + 2
= 5
Matlab, domain ke element 3 ka relationship codomain ke element 5 se hai.
Isi tarah se f(x) mein x ki jagah par 4 rakhne par aur solve karne par,
f(4) = 4 + 2
= 6
Matlab, domain ke element 4 ka relationship codomain ke element 6 se hai.
Isi tarah se f(x) mein x ki jagah par 5 rakhne par aur solve karne par,
f(5) = 5 + 2
= 7
Matlab, domain ke element 5 ka relationship codomain ke element 7 se hai.
Isi tarah se aap domain ke sahbhi elements ka codomain mein image ya relationship nikaal sakta hain.
So, Domain = N
= {1, 2, 3, 4, 5, ...}
Codomain = N
= {1, 2, 3, 4, 5, ...}
Range = N - {1, 2}
= {3, 4, 5, 6, 7, ...}
(Yaha par N mein se 1 aur 2 ko minus kar diya kyuki domain mein koi aise elements nahi hain jo inse related ho).
Ex.2. A function is defined by g : {1, 2, 3} ➝ {-1, -2, -3, -4}, g(x) = -x. Draw its arrow diagram. Also find its domain, codomain and range.
Solution: g : {1, 2, 3} ➝ {-1, -2, -3, -4}
Ab g(x) mein x ki jagah par 1 rakhne par aur solve karne par,
g(1) = -1
Matlab, domain ke element 1 ka relationship codomain ke element -1 se hai.
Isi tarah se g(x) mein x ki jagah par 2 rakhne par aur solve karne par,
g(2) = -2
Matlab, domain ke element 2 ka relationship codomain ke element -2 se hai.
Isi tarah se g(x) mein x ki jagah par 3 rakhne par aur solve karne par,
g(3) = -3
Matlab, domain ke element 3 ka relationship codomain ke element -3 se hai.
So, Domain = {1, 2, 3}
Codomain = {-1, -2, -3, -4}
Range = {-1, -2, -3}
Ex.3. State if the relation {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)} a function? If it is a function, determine its domain and range.
Solution: Relation = {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}.
Chaliye iss relation ka arrow diagram banaate hain:
Aap ye clearly dekh sakte hai ki iss relation ke domain ke har(every) element ka relationship iss set ke codomain ke sirf ek hi element (unique element) se hai. So, ye relation ek function hai. So, iss function ka domain wahi hoga jo iss relation ka domain hai.
Domain of function = {2, 5, 8, 11, 14}
Iss function ka range bhi wahi hoga jo iss relation ka range hai.
Range of function = {1}.
Ex.3. State if the relation {(1, 3), (1, 5), (2, 5)} a function? If it is a function, determine its domain and range.
Solution: Relation = {(1, 3), (1, 5), (2, 5)}.
Chaliye iss relation ka arrow diagram banaate hain:
Ex.5. A function is Defined by f : Z - {0}➝ N, f(x) = x2. State f(-3), f(5) and f(-10). Draw its arrow diagram. Also state its domain, codomain and range.
Solution: f : Z - {0}➝ N, f(x) = x2
Ab f(x) mein x ki jagah par -3 rakhne par aur solve karne par,
f(-3) = (-3)2
= 9
Matlab, domain ke element -3 ka relationship codomain ke element 9 se hai.
Isi tarah se f(x) mein x ki jagah par 5 rakhne par aur solve karne par,
f(5) = (5)2
= 25
Matlab, domain ke element 5 ka relationship codomain ke element 25 se hai.
Isi tarah se f(x) mein x ki jagah par -10 rakhne par aur solve karne par,
f(-10) = (-10)2
= 100
Matlab, domain ke element -10 ka relationship codomain ke element 100 se hai.
Isi tarah se aap domain ke sahbhi elements ka codomain mein image ya relationship nikaal sakta hain.
Domain = Z - {0}
= { ..., -2, -1, 1, 2, ...}
Codomain = N
= {1, 2, 3, 4, ...}
Range = {1, 4, 9, 16, 25, ...}
Mujhe umeed hai aap Function se kafi achchhi tarah se parichit (परिचित) ho gaye honge.
[Important Notes:
1. Kisi function ko kabhi-kabhi sirf uss functuion ka rule bataakar bhi show kiya jaata hai, uss function ka Domain aur Codomain bataane ki koi jaroorat nahi hoti hai. Example ke liye maan lijiye ek function hai, f : R ➝ R, f(x) = 3x2 + 4x - 8. Isko sirf ye likhkar hi show kiya jaa sakta hai, f(x) = 3x2 + 4x - 8, x ∈ R.
2. f(x) ya g(x) ya h(x) etc. ko sirf ek single letter jaise ki 's' ya 't' ya 'y' ya 'z' etc. se bhi show kiya jaata hai. Example ke liye, function f : R ➝ R, f(x) = 3x2 + 4x - 8 mein f(x) ko y bhi likh sakte hain [i.e., y = f(x)]. Aisa karne par ye function aisa ho jaayega, f : R ➝ R, y = 3x2 + 4x - 8.
3. Agar practically kaha jaaye to - " If any quantity depends on the other quantity, then the first quantity is called the function of the other quantity".
Matlab, agar koi cheez kisi doosre cheez par depend karti hai to ye kaha jaata hai ki pehli cheez doosri cheez ka function hai. For example, hamari age (उम्र), time par depend karti hai. Jaise-jaise time aage chalta jaata hai waise-waise hamaari age bhi change hoti jaati hai. So, Mathematics ki language mein hum ye kahenge ki ' Hamaari age, time ka function hai'. Aur isko Mathematics ki language mein aise show karte hain: 'our age = f(time).' Isi tarah se kisi circle ka area uski radius par depend karta hai. Agar kisi circle ki radius kam kar di jaaye to uska area bhi kam ho jaayega aur agar uss circle ki radius jyada kar di jaaye to uska area bhi jyada ho jaayega. So, Mathematics ki language mein hum ye kahenge ki 'Kisi circle ka area, uske radius ka function hai'. Aur isko Mathematics ki language mein aise show karte hain: 'Area of circle = f(radius).' Isi tarah se agar koi quantity 'y', kisi doosri quantity 'x' par depend karta. So, Mathematics ki language mein hum ye kahenge ki 'y, x ka function hai'. Aur isko Mathematics ki language mein aise show karte hain: 'y = f(x)'].
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
- Difference of Sets
- Symmetric Difference of Sets
- What are Intervals ?
- What is Cartesian Product ?
- What is Relation ?
THANKS FOR READING THIS BLOG.
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