🍫 Steps to Write a Set in Set-builder Form

To convert the given set in set—builder form, we use the following steps
Step I. Describe the elements of the set by using a symbol x or any other symbol y, z etc.
Step II. Write the symbol colon ‘:’.
Step III. After the sign of colon, write the characteristic property possessed by the elements of the set.
Step IV. Enclose the whole description within braces i.e., { }.

Representation of a Statement in Both Forms

Statement	Roster form	Set-builder form
The set of all natural numbers between 10 and 14.	{11, 12, 13}	{x:x ∈ℕ, 10<x<14}
The set of all schools in Ottawa beginning with letter A.	{Abraar School, Académie Réveil, Ashbury College, Astolot Educational Centre, …}	{x:x is name of school Delhi beginning with letter A}
The set of all distinct letters used in the word ‘Friend’.	{F,r,i,e,n,d}	{x:x is the distinct letters used in the word ‘Friend’}

Example 1:Worked out a Problem
Write the following set, A={14,21,28,35,42,…,98} in set—builder form.

Step I.	Describe the elements of the set by using a symbol. Let x represent the elements of given set.
Step II.	Write the symbol colon. Write the symbol ‘:’ after x.
Step III.	Find the characteristic property possessed by the elements of the set. Given numbers are all natural numbers greater than 7 which are multiples of 7 and less than 100.
Step IV.	Enclose the whole description within braces. Thus, A={x:x is a set of natural numbers greater than 7 which are multiples of 7 and less than 100}.

Example 2:Using Set-Builder Notation
a) Write set B={1,2,3,4,5} in set-builder notation.
b) Write. in words, how you would read set B in set-builder notation.
Solution:
a) Because set B consists of the natural numbers less than 6. we write

B={x|x∈ℕ and x<6}
Another acceptable answer is B={x|x∈ℕ and x≤5}.
b) Set B is the set of all elements x such that x is a natural number and x is less than 6.

Example 3:Roster Form to Set-Builder Notation
a) Write set C={North America, South America, Europe, Asia, Australia. Africa. Antarctica} in set-builder notation.
b) Write in words how you would read set C in set-builder notation.
Solution:
a) C={x|x is a continent}.
b) Set C is the set of all elements x such that x is a continent.

Ex4. Write the set D={1,4,9,16,25,…}in set—builder form.
✍ Solution:We may write the set D as D={x:x is the square of a natural number} Alternatively, we can write D={x:x=n², where n∈ℕ}

Ex5. Write the set {½,⅔,¾,⅘,⅚,⁶⁄₇} in set—builder form.
✍ Solution:We see that each member in the given set has the numerator one less than the denominator. Also, the numerator begin from 1 and do not exceed 6. Hence, in the set—builder form the given set is
{x:x =ⁿ/_(n+1), where n is a natural number and 1≤n≤6}

Ex6. Write set E={3,6,9,12,15} in set-builder form.
Sol. E={x:x is a natural number multiple of 3 and x<18}. Ex7. Write set F={1,4,9,…,100} in set—builder form.
Sol. F={x:x =n², n∈ℕ and n<11} Ex8. Write set G={½,⅖,³⁄₁₀,⁴⁄₁₇,⁵⁄₂₆,⁶⁄₃₇,⁷⁄₅₀} in set-builder form.
Sol. G={x:x =n/(n²+1), n∈ℕ and n≤7}
💪 How to Insert the appropriate Symbol either an Element or not an element to a Set?
Ex9. Here are some further illustrations of set-builder notation.
1. {n:n is a prime number}={2,3,5,7,11,13,17,…}
2. {n∈ℕ:n is prime}={2,3,5,7,11,13,17,…}
3. {n²:n∈ℤ}={0,1,4,9,16,25,…}
4. {x∈ℝ:x²-2=0}={√2,- √2}
5. {x∈ℤ:x²-2=0}=Ø
6. {x∈ℤ:|x|<4}={-3,-2,-1,0,1,2,3} 7. {2x∶x∈ℤ,|x|<4}={-6,-4,-2,0,2,4,6} 8. {x∈ℤ:|2x|<4}={-1,0,1}

These last three examples highlight a conflict of notation that we must always be alert to. The expression |X| means absolute value if X is a number and cardinality if X is a set. The distinction should always be clear from context. Consider {x∈ℤ:|x|<4} in Ex9 (6) above. Here x∈ℤ, so x is a number (not a set), and thus the bars in |x| must mean absolute value, not cardinality. On the other hand, suppose A={{1,2},{3,4,5,6},{7}} and B={X∈A:|X|<3}. The elements of A are sets (not numbers), so the |X| in the expression for B must mean cardinality. Therefore B={{1,2},{7}}.