Sets · Lesson 1 of 6
Understanding and Representing Sets
“Learn how a clear membership rule turns a collection into a set and how to describe that set precisely.”
• Decide whether a collection has a well-defined membership rule. • Read and use membership notation and familiar number-set symbols. • Represent sets in roster and set-builder forms without losing restrictions. • Translate between descriptions, equations, and the elements they identify. • Explain why order and repeated entries do not change a set.
Why mathematicians use sets
Suppose you want to describe every natural number that divides 42. You could write a sentence, list the numbers, or give a rule. These descriptions serve the same purpose: they identify a collection whose membership can be checked. Sets provide a precise language for working with such collections, and later help us describe functions, geometric objects, sequences, and probability.
A collection becomes a set when the question “Does this object belong?” has a definite answer. The answer may require work, but it must not depend on personal taste. For instance, deciding whether a number is a prime factor of 210 may require factorisation. That difficulty does not make the collection unclear: 210 = 2 × 3 × 5 × 7, so the prime factors are exactly 2, 3, 5, and 7.
A set is a well-defined collection of objects. A stated rule or context determines whether each object belongs to the collection.
The vowels of the English alphabet, odd natural numbers less than 10, and the solutions of x² − 5x + 6 = 0 are sets. Each has an objective membership test. By contrast, “the five most talented writers” is unclear without an agreed criterion for talent. Different readers can choose different members while all following their own interpretation.
Context also matters. “Students in my class” identifies a definite collection when the class and time are fixed. A mathematical description need not remain unchanged forever to describe a set; it needs a clear meaning in the situation being discussed. The same applies to rivers in a specified region or members of a named team.
Problem
Compare the months whose English names begin with J with the three most enjoyable months of the year.
- 1.Test month names against the first rule. January, June, and July qualify, so the collection has definite members.
- 2.“Most enjoyable” gives no shared test. A person who enjoys winter and a person who enjoys summer can disagree without either violating the wording.
- 3.The first description defines a set. The second needs an objective selection rule before it defines a set.
Elements and membership
Once the collection is well-defined, we need a way to name it and refer to its members. We usually use capital letters for sets and lower-case letters for general elements. These conventions make an expression easier to read, but the meaning comes from the definitions we give.
An object belonging to a set is called an element or member of that set. The terms object, element, and member refer to the objects collected by the set.
If V is the set of English vowels, a belongs to V while b does not. The symbol ∈ means “is an element of”; the symbol ∉ means “is not an element of”. Thus a ∈ V and b ∉ V. Read the whole expression aloud: the letter on the left is being tested against the set on the right.
Some number sets have standard names. In this chapter natural numbers begin at 1. Integers include zero and negative whole numbers. Rational numbers are numbers expressible as a fraction of integers with a nonzero denominator. Real numbers include both rational and irrational numbers. These meanings will become especially useful when the same equation is solved in different domains.
| Symbol | Meaning | Examples of members |
|---|---|---|
| N | Natural numbers: 1, 2, 3, … | 1, 7, 100 |
| Z | Integers: …, −2, −1, 0, 1, 2, … | −4, 0, 8 |
| Q | Rational numbers | −3, 0, 2/5 |
| R | Real numbers | −3, 2/5, √2, π |
| Z⁺ | Positive integers | 1, 2, 3 |
| Q⁺ | Positive rational numbers | 1/2, 3, 7/4 |
| R⁺ | Positive real numbers | 1/2, √2, 5 |
A divisor and a prime divisor are different descriptions. Although 15 divides 30, it is not prime. If P is the set of prime factors of 30, then 3 ∈ P but 15 ∉ P. Check every part of the membership rule.
Roster form
Roster form shows a set by displaying its elements inside braces. It is useful when the members can be listed clearly. The braces tell us that the entries form a collection rather than an ordered calculation or sequence of instructions.
List the distinct elements, separate them with commas, and enclose the list in braces { }. For example, the positive even integers less than 7 form {2, 4, 6}.
The set of positive divisors of 42 is {1, 2, 3, 6, 7, 14, 21, 42}. One reliable way to obtain the complete list is to find factor pairs: 1 × 42, 2 × 21, 3 × 14, and 6 × 7. Each member passes the divisibility test, and pairing avoids forgetting the larger divisors.
Order does not affect membership. The lists {1, 2, 3} and {3, 1, 2} describe the same set. Repeating an entry also changes nothing: {1, 1, 2, 3, 3} still has only the distinct members 1, 2, and 3. A set records which objects occur, not how often they occur.
Problem
Write the set of letters used in the word SCHOOL.
- 1.Inspect the letters S, C, H, O, O, L.
- 2.Keep O once because the object being collected is a letter, not a particular occurrence of that letter.
- 3.The set is {S, C, H, O, L}. Any rearrangement of these five distinct letters describes the same set.
An ellipsis, written …, can indicate a continuing pattern: {1, 3, 5, 7, …} describes the odd natural numbers when that pattern has been stated. It may also abbreviate a finite list, such as {1, 2, …, 20}. The dots do not automatically mean “infinite”. They depend on the description and any final endpoint.
A short list does not determine a unique continuation by itself. When using dots, state the intended pattern. For a set with complicated or densely packed members, a membership rule is usually clearer than an attempted list.
Set-builder form
Instead of naming the elements individually, set-builder form names the rule that every member must satisfy. This is especially useful for an infinite collection or a collection described by an equation. A good rule tells us both what kind of object is allowed and which condition it must meet.
In V = {x : x is a vowel in the English alphabet}, the braces mean “the set of all”, the colon means “such that”, and x is a placeholder for a possible member. You could replace x throughout by y without changing the set. It is the condition after the colon, not the choice of letter, that selects the elements.
Here the domain is N, so fractions and decimals are excluded even if they lie between 3 and 10. The inequalities exclude the endpoints. Testing the natural numbers leaves A = {4, 5, 6, 7, 8, 9}. If the domain were R instead, the same inequalities would allow infinitely many real numbers between those endpoints.
Problem
Write the solution set of x² + x − 2 = 0 in roster form.
- 1.We seek numbers making the expression zero, so first factor it: x² + x − 2 = (x − 1)(x + 2).
- 2.A product of real numbers is zero only if at least one factor is zero. Thus x − 1 = 0 or x + 2 = 0.
- 3.The candidates are x = 1 and x = −2. Substitution gives 1 + 1 − 2 = 0 and 4 − 2 − 2 = 0.
- 4.Both qualify. The solution set is {1, −2}. Writing −2 first would not change it.
Problem
Convert {x : x is a positive integer and x² < 40} into roster form.
- 1.Since x is positive, start with 1 and increase through the positive integers.
- 2.The squares of 1 through 6 are 1, 4, 9, 16, 25, and 36, all less than 40.
- 3.The next square is 7² = 49, already too large. Every larger positive integer has an even larger square.
- 4.The set is {1, 2, 3, 4, 5, 6}. Zero and negative integers are excluded by the domain, even though some would satisfy the inequality.
Converting a pattern into a rule
To move from a roster to set-builder form, describe exactly the listed objects and any intended continuation. Check that your rule includes every intended member and excludes everything else. A visually plausible pattern is not enough if you forget its bounds.
Problem
Write {1, 4, 9, 16, 25, …}, stated to continue through all natural-number squares, in set-builder form.
- 1.Identify each member as n² for a natural number n.
- 2.The natural-number condition makes n begin at 1 and continue without a final value.
- 3.The set is {x : x = n² for some n ∈ N}. Equivalently, it is the set of squares of natural numbers.
- 4.A check with n = 1, 2, and 3 produces 1, 4, and 9. The rule also correctly continues with 36 and 49.
Problem
Write {1/2, 2/3, 3/4, 4/5, 5/6, 6/7} in set-builder form.
- 1.The numerator is a natural number n and the denominator is n + 1.
- 2.The displayed list begins at n = 1 and ends at n = 6. Those restrictions must be retained.
- 3.Use {x : x = n/(n + 1), n ∈ N, 1 ≤ n ≤ 6}.
- 4.Without the upper bound, the rule would also include 7/8 and further fractions, so it would describe a different set.
Problem
Match {P, R, I, N, C, A, L}, {0}, {1, 2, 3, 6, 9, 18}, and {3, −3} to suitable verbal or algebraic rules.
- 1.The first set is the distinct letters in PRINCIPAL. Its repeated P and I contribute no new elements.
- 2.The equation x + 1 = 1 has the single solution 0, so its integer solution set is {0}.
- 3.Positive divisors of 18 are found from 1 × 18, 2 × 9, and 3 × 6, giving the third set.
- 4.The equation x² − 9 = 0 factors as (x − 3)(x + 3) = 0, giving the fourth set.
- 5.Matching requires equality of the resulting members, rather than similarity of how the descriptions look.
The first exercise boundary brings together well-defined collections, membership symbols, and conversion between the two forms. When approaching those exercise questions, begin by underlining the domain and each restriction. The explanations above supply the concepts; the exercise solutions will appear in the separate solved-exercises area.
Check your understanding
These questions test whether you can interpret a rule and preserve its meaning when the notation changes. Choose an answer before opening the explanation. Pay particular attention to domain restrictions and to the difference between a repeated entry and a new member.
Quiz
Which description defines a set without adding a subjective selection rule?
Month names give a definite membership test. Exciting, nicest, and best need an agreed criterion.
If P = {2, 3, 5}, which statement is true?
Only 3 is one of the listed members. Being divisible by a listed member does not make 15 a member.
Which roster represents {x : x ∈ Z and −1 ≤ x < 3}?
Include −1 because of ≤, exclude 3 because of <, and include the integer 0.
Which rule describes exactly {3, 6, 9, 12}?
The multiplier must be 1, 2, 3, or 4. Omitting the bound includes 15 and many more members.
What happens when {a, b, a, c} is rewritten as {c, a, b}?
Both expressions contain exactly a, b, and c. Repetition and order do not change membership.
Which is the solution set of x² = 9 when x ∈ Z?
Both −3 and 3 are integers and have square 9. The domain does not exclude negative integers.
Key Takeaways
• A set has a definite membership test; the test can require calculation. • Use ∈ and ∉ to state whether an object belongs to a set. • Roster form lists distinct members; order and repetition do not change the set. • Set-builder form needs an accurate domain, condition, and any bounds. • Translate by checking actual members, not by comparing the appearance of descriptions.
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Next · Lesson 2
Types of Sets and Equality