What are Intervals ?
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Friends aaj hum baat karne waale hain intervals ki. Iss article mein me aapko bataane waala hoon intervals ke baare mein. Iss article mein aapko iss sawaal ka jawaab milega - What are Intervals? So friends, bane rahiye mere saath iss article ke end tak...
Let's Begin...
Friends, intervals real numbers (R) ke subsets hote hain. Maan lijiye aapko ek set banana hai jiske elements 2 aur 5 ke beech ke sabhi numbers hain. To iss set ko aap kaise banaoge? Iss set ko aap Roster form mein nahi likh sakte. Iss set ko aap sirf Set-builder form mein likh sakte ho. Chaliye iss set ko Set-builder form mein likhte hain - {x: 2 < x < 5}. Iss set ko hum interval bhi kehte hain. Iss set ya interval ko chaliye number line par show karte hain-
Aap dekh sakte hain ki iss diagram mein 2 aur 5 ke beech ka part coloured hai. Ye coloured part 2 aur 5 ke beech ke sabhi real numbers ko show kar raha hai jo ki iss set ya interval ke elements hain aur aap ye bhi dekh sakte hain ki 2 par aur 5 par hollow circle ( ൦ ) hai. Jiska matlab hai ki iss set ya interval mein naa hi 2 ko lena hai aur naa hi 5 ko lena hai.
To aapne dekha friends intervals, number line ka ek part hota hai.
Chaliye ek aur example lete hain interval se related. Maan lijiye aapko ek set banana hai jiske elements wo real numbers ho jo -2 aur 0 ko shaamil karte hue (include karke), -2 aur 0 ke beech ke hain. Iss set ko aap Set-builder form mein likh sakte hain - {x: -2 ≤ x ≤ 0}. Iss set ke elements real numbers (number line) ka part hain isiliye ye set ek interval hai. Iss set ya interval ko chaliye number line par show karte hain-
Aap dekh sakte hain ki iss diagram mein -2 aur 0 ke beech ka part coloured hai. Ye coloured part -2 aur 0 ke beech ke sabhi real numbers ko show kar raha hai jo ki iss set ya interval ke elements hain aur aap ye bhi dekh sakte hain ki -2 par aur 0 par hollow circle ( ൦ ) hai. Jiska matlab hai ki iss set ya interval mein -2 ko lena hai aur 5 ko bhi lena hai.
To friends, aapko pata chal gaya hoga ki intervals kya hote hain. Friends, intervals ko symbolically likhne ke do(two) tarike/methods/ways hain-
(i) Set ke form mein
(ii) Brackets ka use karke
Friends, hame pehle method (Set ke form mein) pata hai. Upar ke dono examples mein maine intervals ko set ke form mein hi likha hai.
Friends, intervals ko likhne ka jo doosra method hai (Brackets ka use karke), wo bahut jyada popular method hai aur sabhi log intervals ko iss method mein hi likhte hain.
To chaliye jaante hain ki intervals ko likhne ka doosra tarika/method hai...
Writing Intervals Using Brackets
Friends interval mein do(two) numbers ke beech ke elements hote hain, unn numbers ko extreme points kehte hain aur unko comma ( , ) ka use karke alag-alag(separate) karke likha jaata hai aur unn dono numbers ko small bracket ya square bracket mein close karke likha jata hai according to situation. Intervals mein agar left wala extreme point included hai to iske aage '[' wala bracket lagaate hain aur agar yeh included nahi hai to iske aage '(' ya ']' wala bracket lagaate hain. Isi tarah se intervals mein agar right wala extreme point included ho to iske peeche ']' wala bracket lagaate hain aur agar ye included nahi hai to iske peeche ')' ya '[' waala bracket lagaate hain. Example ke liye ye interval dekhiye- [3, 7). Iss interval mein 3 include hai and 3 aur 7 ke beech jitne bhi real numbers hain wo bhi include hain but 7 include nahi hai.
Types of Intervals
Maan lijiye ki a aur b real numbers hain aur a < b tab,
(i) [a, b] - Iss interval ko closed interval kehte hain kyuki isme dono extreme points included hain. Isko set ke form mein aise show karte hain- {x: a ≤ x ≤ b}. Isko number line par show karne par kuch aisa dikhega-
(ii) (a, b) ya ]a, b[ - Iss interval ko open interval kehte hain kyuki isme dono extreme points included hain. Isko set ke form mein aise show karte hain- {x: a < x < b}. Isko number line par show karne par kuch aisa dikhega-
(iii) (a, b] ya ]a, b] - Iss interval ko semi-open (ya) half-open (ya) semi-closed (ya) half-closed kehte hain kyuki isme dono extreme points mein se ek extreme point included hai. Isko set ke form mein aise show karte hain- {x: a < x ≤ b}. Isko number line par show karne par kuch aisa dikhega-
(iv) [a, b) ya [a, b[ - Iss interval ko semi-open (ya) half-open (ya) semi-closed (ya) half-closed kehte hain kyuki isme dono extreme points mein se ek extreme point included hai. Isko set ke form mein aise show karte hain- {x: a ≤ x < b}. Isko number line par show karne par kuch aisa dikhega-
(v) (a, ∞) ya (a, ∞[ ya ]a, ∞) ya ]a, ∞[ - Waise to ye ek open interval hai lekin isko doosre naam(name) se bhi jaana jaata hai- 'open right ray'. Iska naam(name) open right ray iss wajah se pada kyuki jab isko number line par show karte hain to ek ray jaisa shape ban jaata hai jiska arrow right hand side ki taraf hota hai aur isme left waala extreme point include nahi kiya jaata hai. Isko set ke form mein aise show karte hain- {x: a < x}. Isko number line par show karne par kuch aisa dikhega-
(vi) [a, ∞) ya [a, ∞[ - Ye ek half-open interval hai lekin isko doosre naam(name) se bhi jaana jaata hai- 'closed right ray'. Iska naam(name) closed right ray iss wajah se pada kyuki jab isko number line par show karte hain to ek ray jaisa shape ban jaata hai jiska arrow right hand side ki taraf hota hai aur isme left waala extreme point include kiya jaata hai. Isko set ke form mein aise show karte hain-
{x: a ≤ x}. Isko number line par show karne par kuch aisa dikhega-
(vii) (-∞, b) ya (-∞, b[ ya ]-∞, b) ya ]-∞, b[ - Ye ek open interval hai lekin isko doosre naam(name) se bhi jaana jaata hai- 'open left ray'. Iska naam(name) open left ray iss wajah se pada kyuki jab isko number line par show karte hain to ek ray jaisa shape ban jaata hai jiska arrow left hand side ki taraf hota hai aur isme right waala extreme point include nahi kiya jaata hai. Isko set ke form mein aise show karte hain- {x:x < b}. Isko number line par show karne par kuch aisa dikhega-
(viii) (-∞, b] ya ]-∞, b] - Ye ek half-open interval hai lekin isko doosre naam(name) se bhi jaana jaata hai- 'closed left ray'. Iska naam(name) closed left ray iss wajah se pada kyuki jab isko number line par show karte hain to ek ray jaisa shape ban jaata hai jiska arrow left hand side ki taraf hota hai aur isme left waala extreme point include kiya jaata hai. Isko set ke form mein aise show karte hain-
{x:x ≤ b}. Isko number line par show karne par kuch aisa dikhega-
(ix) (-∞, ∞) ya ]-∞, ∞[ ya R - Ye ek open interval hai. Iss interval mein -∞ aur ∞ ke beech ke sabhi real numbers aate hain. Kehne ka matlab ye hai ki iss interval mein sabhi real numbers aate hain. Matlab ye interval poore real number ke set (R) ko ya poore number line ko show karta hai. Isko number line par show karne par kuch aisa dikhega-
[Note:
1. ∞ aur -∞ ki taraf interval hamesha open hota hai.
2. Kisi interval [a, b] ya (a, b) ya (a, b] ya [a, b] ki length hoti hai: b - a].
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 are Intervals?
Friends, intervals real numbers (R) ke subsets hote hain. Maan lijiye aapko ek set banana hai jiske elements 2 aur 5 ke beech ke sabhi numbers hain. To iss set ko aap kaise banaoge? Iss set ko aap Roster form mein nahi likh sakte. Iss set ko aap sirf Set-builder form mein likh sakte ho. Chaliye iss set ko Set-builder form mein likhte hain - {x: 2 < x < 5}. Iss set ko hum interval bhi kehte hain. Iss set ya interval ko chaliye number line par show karte hain-
Aap dekh sakte hain ki iss diagram mein 2 aur 5 ke beech ka part coloured hai. Ye coloured part 2 aur 5 ke beech ke sabhi real numbers ko show kar raha hai jo ki iss set ya interval ke elements hain aur aap ye bhi dekh sakte hain ki 2 par aur 5 par hollow circle ( ൦ ) hai. Jiska matlab hai ki iss set ya interval mein naa hi 2 ko lena hai aur naa hi 5 ko lena hai.
To aapne dekha friends intervals, number line ka ek part hota hai.
Chaliye ek aur example lete hain interval se related. Maan lijiye aapko ek set banana hai jiske elements wo real numbers ho jo -2 aur 0 ko shaamil karte hue (include karke), -2 aur 0 ke beech ke hain. Iss set ko aap Set-builder form mein likh sakte hain - {x: -2 ≤ x ≤ 0}. Iss set ke elements real numbers (number line) ka part hain isiliye ye set ek interval hai. Iss set ya interval ko chaliye number line par show karte hain-
Aap dekh sakte hain ki iss diagram mein -2 aur 0 ke beech ka part coloured hai. Ye coloured part -2 aur 0 ke beech ke sabhi real numbers ko show kar raha hai jo ki iss set ya interval ke elements hain aur aap ye bhi dekh sakte hain ki -2 par aur 0 par hollow circle ( ൦ ) hai. Jiska matlab hai ki iss set ya interval mein -2 ko lena hai aur 5 ko bhi lena hai.
To friends, aapko pata chal gaya hoga ki intervals kya hote hain. Friends, intervals ko symbolically likhne ke do(two) tarike/methods/ways hain-
(i) Set ke form mein
(ii) Brackets ka use karke
Friends, hame pehle method (Set ke form mein) pata hai. Upar ke dono examples mein maine intervals ko set ke form mein hi likha hai.
Friends, intervals ko likhne ka jo doosra method hai (Brackets ka use karke), wo bahut jyada popular method hai aur sabhi log intervals ko iss method mein hi likhte hain.
To chaliye jaante hain ki intervals ko likhne ka doosra tarika/method hai...
Writing Intervals Using Brackets
Friends interval mein do(two) numbers ke beech ke elements hote hain, unn numbers ko extreme points kehte hain aur unko comma ( , ) ka use karke alag-alag(separate) karke likha jaata hai aur unn dono numbers ko small bracket ya square bracket mein close karke likha jata hai according to situation. Intervals mein agar left wala extreme point included hai to iske aage '[' wala bracket lagaate hain aur agar yeh included nahi hai to iske aage '(' ya ']' wala bracket lagaate hain. Isi tarah se intervals mein agar right wala extreme point included ho to iske peeche ']' wala bracket lagaate hain aur agar ye included nahi hai to iske peeche ')' ya '[' waala bracket lagaate hain. Example ke liye ye interval dekhiye- [3, 7). Iss interval mein 3 include hai and 3 aur 7 ke beech jitne bhi real numbers hain wo bhi include hain but 7 include nahi hai.
Types of Intervals
Maan lijiye ki a aur b real numbers hain aur a < b tab,
(i) [a, b] - Iss interval ko closed interval kehte hain kyuki isme dono extreme points included hain. Isko set ke form mein aise show karte hain- {x: a ≤ x ≤ b}. Isko number line par show karne par kuch aisa dikhega-
(ii) (a, b) ya ]a, b[ - Iss interval ko open interval kehte hain kyuki isme dono extreme points included hain. Isko set ke form mein aise show karte hain- {x: a < x < b}. Isko number line par show karne par kuch aisa dikhega-
(iii) (a, b] ya ]a, b] - Iss interval ko semi-open (ya) half-open (ya) semi-closed (ya) half-closed kehte hain kyuki isme dono extreme points mein se ek extreme point included hai. Isko set ke form mein aise show karte hain- {x: a < x ≤ b}. Isko number line par show karne par kuch aisa dikhega-
(iv) [a, b) ya [a, b[ - Iss interval ko semi-open (ya) half-open (ya) semi-closed (ya) half-closed kehte hain kyuki isme dono extreme points mein se ek extreme point included hai. Isko set ke form mein aise show karte hain- {x: a ≤ x < b}. Isko number line par show karne par kuch aisa dikhega-
(v) (a, ∞) ya (a, ∞[ ya ]a, ∞) ya ]a, ∞[ - Waise to ye ek open interval hai lekin isko doosre naam(name) se bhi jaana jaata hai- 'open right ray'. Iska naam(name) open right ray iss wajah se pada kyuki jab isko number line par show karte hain to ek ray jaisa shape ban jaata hai jiska arrow right hand side ki taraf hota hai aur isme left waala extreme point include nahi kiya jaata hai. Isko set ke form mein aise show karte hain- {x: a < x}. Isko number line par show karne par kuch aisa dikhega-
(vi) [a, ∞) ya [a, ∞[ - Ye ek half-open interval hai lekin isko doosre naam(name) se bhi jaana jaata hai- 'closed right ray'. Iska naam(name) closed right ray iss wajah se pada kyuki jab isko number line par show karte hain to ek ray jaisa shape ban jaata hai jiska arrow right hand side ki taraf hota hai aur isme left waala extreme point include kiya jaata hai. Isko set ke form mein aise show karte hain-
{x: a ≤ x}. Isko number line par show karne par kuch aisa dikhega-
(vii) (-∞, b) ya (-∞, b[ ya ]-∞, b) ya ]-∞, b[ - Ye ek open interval hai lekin isko doosre naam(name) se bhi jaana jaata hai- 'open left ray'. Iska naam(name) open left ray iss wajah se pada kyuki jab isko number line par show karte hain to ek ray jaisa shape ban jaata hai jiska arrow left hand side ki taraf hota hai aur isme right waala extreme point include nahi kiya jaata hai. Isko set ke form mein aise show karte hain- {x:x < b}. Isko number line par show karne par kuch aisa dikhega-
(viii) (-∞, b] ya ]-∞, b] - Ye ek half-open interval hai lekin isko doosre naam(name) se bhi jaana jaata hai- 'closed left ray'. Iska naam(name) closed left ray iss wajah se pada kyuki jab isko number line par show karte hain to ek ray jaisa shape ban jaata hai jiska arrow left hand side ki taraf hota hai aur isme left waala extreme point include kiya jaata hai. Isko set ke form mein aise show karte hain-
{x:x ≤ b}. Isko number line par show karne par kuch aisa dikhega-
(ix) (-∞, ∞) ya ]-∞, ∞[ ya R - Ye ek open interval hai. Iss interval mein -∞ aur ∞ ke beech ke sabhi real numbers aate hain. Kehne ka matlab ye hai ki iss interval mein sabhi real numbers aate hain. Matlab ye interval poore real number ke set (R) ko ya poore number line ko show karta hai. Isko number line par show karne par kuch aisa dikhega-
[Note:
1. ∞ aur -∞ ki taraf interval hamesha open hota hai.
2. Kisi interval [a, b] ya (a, b) ya (a, b] ya [a, b] ki length hoti hai: b - a].
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
THANKS FOR READING THIS BLOG.
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