There are two methods of representing a set :
(i) Roster or tabular form
(ii) Set-builder form
`\color{green} {(i)" Roster or tabular form :"}`
`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT" ` In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces `{ }.`
E.g. the set of all even positive integers less than 7 is described in roster form as {2, 4, 6}.
Some more examples of representing a set in roster form are given below :
`(a)` The set of all natural numbers which divide `42` is `{1, 2, 3, 6, 7, 14, 21, 42}.`
`(b)` The set of all vowels in the English alphabet is `{a, e, i, o, u}.`
`(c)` The set of odd natural numbers is represented by `{1, 3, 5, . . .}.` The dots tell us that the list of odd numbers continue indefinitely.
`color{green} {"Note"}` It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct.
For example, the set of letters forming the word `"‘SCHOOL’"` is `{ S, C, H, O, L}` or `{H, O, L, C, S}.` Here, the order of listing elements has no relevance.
`\color{green} {"(ii) Set-builder form :"}`
`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT" ` In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
`\color{green} ✍️ \color{green} \mathbf( Method :)` - We describe the element of the set by using a symbol `x,` which is followed by a colon `"“ : ”."`
- After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces.
E.g. In the set `{a, e, i, o, u}`, is a vowel in the English alphabet. Denoting this set by V, we can write as `V = {x : x \ \ "is a vowel in English alphabet}"` in set builder form.
E.g. `A = {x : x \ \ "is a natural number and" \ \ 3 < x < 10}` is read as “the set of all `x` such that `x` is a natural number and `x` lies between `3` and `10.`
Hence, the numbers `4, 5, 6, 7, 8` and `9` are the elements of the set `A.`
If we denote the sets described in (a), (b) and (c) above in roster form by A, B, C, respectively, then A, B, C can also be represented in set-builder form as follows:
`A= {x : x " is a natural number which divides" 42}`
`B= {y : y "is a vowel in the English alphabet" }`
`C= {z : z "is an odd natural number" }`
There are two methods of representing a set :
(i) Roster or tabular form
(ii) Set-builder form
`\color{green} {(i)" Roster or tabular form :"}`
`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT" ` In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces `{ }.`
E.g. the set of all even positive integers less than 7 is described in roster form as {2, 4, 6}.
Some more examples of representing a set in roster form are given below :
`(a)` The set of all natural numbers which divide `42` is `{1, 2, 3, 6, 7, 14, 21, 42}.`
`(b)` The set of all vowels in the English alphabet is `{a, e, i, o, u}.`
`(c)` The set of odd natural numbers is represented by `{1, 3, 5, . . .}.` The dots tell us that the list of odd numbers continue indefinitely.
`color{green} {"Note"}` It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct.
For example, the set of letters forming the word `"‘SCHOOL’"` is `{ S, C, H, O, L}` or `{H, O, L, C, S}.` Here, the order of listing elements has no relevance.
`\color{green} {"(ii) Set-builder form :"}`
`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT" ` In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
`\color{green} ✍️ \color{green} \mathbf( Method :)` - We describe the element of the set by using a symbol `x,` which is followed by a colon `"“ : ”."`
- After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces.
E.g. In the set `{a, e, i, o, u}`, is a vowel in the English alphabet. Denoting this set by V, we can write as `V = {x : x \ \ "is a vowel in English alphabet}"` in set builder form.
E.g. `A = {x : x \ \ "is a natural number and" \ \ 3 < x < 10}` is read as “the set of all `x` such that `x` is a natural number and `x` lies between `3` and `10.`
Hence, the numbers `4, 5, 6, 7, 8` and `9` are the elements of the set `A.`
If we denote the sets described in (a), (b) and (c) above in roster form by A, B, C, respectively, then A, B, C can also be represented in set-builder form as follows:
`A= {x : x " is a natural number which divides" 42}`
`B= {y : y "is a vowel in the English alphabet" }`
`C= {z : z "is an odd natural number" }`
