How to Write the Elements of a Set from Rules or Patterns - dummies

# How to Write the Elements of a Set from Rules or Patterns

You can describe the elements in a set in several different ways, but you usually want to choose the method that’s quickest and most efficient and/or clearest to the reader. The two main methods for describing a set are roster and rule (or set-builder).

A roster is a list of the elements in a set. When the set doesn’t include many elements, then this description works fine. If the set contains a lot of elements, you can use an ellipsis ( . . . ) if the pattern is obvious (a nasty word in mathematics). A rule works well when you find lots and lots of elements in the set.

## Sample questions

1. Use roster and rule notation to describe the set F, which consists of all the positive multiples of 5 that are less than 50.

Roster notation: F = {5, 10, 15, 20, 25, 30, 35, 40, 45}; rule notation:

You read the rule as “Set F consists of all x’s such that x is equal to five times n, where n is an integer and n is a number between zero and ten.”

The symbol Z stands for integers. Of course, you don’t have to write the elements in the set in order. You can just as easily write F = {10, 20, 30, 40, 45, 35, 25, 15, 5}.

2. Write the elements of set C in roster form if C = {x | x = a2 and x = b3, where 0 < a, b < 30}.

C= {1, 64, 729}. The rule for C is that x has to be a perfect square and a perfect cube. The bases of x (a and b) are positive numbers less than 30.

The best way to approach this problem is to find all the squares of the numbers from 1 to 30 and then determine which are cubes: C = {1, 64, 729}. You get these elements because 1 = 12 = 13, 64 = 82 = 43, and 729 = 272 = 93.

## Practice questions

1. Write set A using roster notation if A = {x | x is odd, x = 7n, 0 < x < 70}.

2. Write set C using a rule if C = {11, 21, 31, 41, 51, 61}.

3. Write set D using a rule if D = {1, 5, 9, 13, 17, 21, … }.

Following are answers to the practice questions:

1. Write set A using roster notation if A = {x | x is odd, x = 7n, 0 < x < 70}. The answer is {7, 21, 35, 49, 63}.

According to the rule, you want numbers that are odd, multiples of 7, and between 0 and 70.

2. Write set C using a rule if C = {11, 21, 31, 41, 51, 61}. The answer is

Another way of writing the rule is

Notice that in this second rule, the values of n are different — they begin and end in different places — and the constant 11 is different. What’s alike in these two rules is that the n is multiplied by 10, keeping the terms 10 units apart.

3. Write set D using a rule if D = {1, 5, 9, 13, 17, 21, … }. The answer is

The rule allows the set to be infinite — the number of terms has no end.