How do sets help you organise information without making a mess?
You already use the idea of a set whenever you group things by a clear rule: apps on your phone, students in your class, multiples of 3, vowels, prime numbers. In maths, a set is just a group where membership is unambiguous. If something belongs, it is in. If it does not, it is out.
A set is a collection treated as one whole group, and each object inside it is called an element.
You usually label sets with capital letters such as A, B or C, and you write the elements inside curly brackets. If , then and . The symbol tells you an item belongs to the set, while says it does not.
A weird thing at first: sets do not behave like shopping lists. The order changes nothing, so and mean exactly the same set. Repeats also do nothing, so if the only ages in a class are 15 and 16, the set of ages is just , even if lots of students share those ages.
Do not use repeated entries to show how often something appears. A set only records membership, not frequency.
Another useful idea is cardinality, which is just the number of different elements in a set.
The cardinality of a set , written , is how many elements are in .
If , then . Some sets stop, so they are finite. Others keep going forever, so they are infinite.
Method: find the cardinality of a listed set
- Write the set clearly without duplicates.
- Count each distinct element once.
- State the answer as .
Worked example
Let be the set of positive integers less than 30 that are multiples of 4.
- List the multiples of 4 below 30: .
- Write the set: .
- Count the elements: there are 7 of them.
- So .
This matters later in probability and data because Venn diagram questions often ask you how many items land in each region. If your set writing is sloppy, your counting goes wrong fast.
How can you describe the same set in different ways?
A set can be written in three main styles, and exam questions love making you switch between them. You can describe it in words, you can list the elements directly, or you can give a rule that generates the elements.
Starting with words is often easiest because it tells you the idea first. For example, you could describe as the set of vowels in the English alphabet. That only works if your wording is precise enough that everyone would choose the same elements.
If two students could reasonably disagree about whether something belongs, your description is too vague.
Listing elements is called roster notation. For the vowel set, you would write . For number sets, you can use when the pattern is obvious, such as .
Only use when the pattern is unmistakable. is clear because it keeps following the even-number pattern.
When listing every element is awkward or impossible, set-builder notation is better.
Set-builder notation describes a set using a variable and a condition, in the form .
You read the vertical bar as “such that”. So means all natural numbers from 1 to 9 inclusive.
Method: turn words into set-builder notation
- Choose a variable, usually .
- State what kind of numbers or objects can be.
- Add the condition that selects the correct elements.
- Put everything inside curly brackets.
Worked example
Write “integers from 2 to 7 inclusive” in set-builder notation.
- Use the variable .
- Restrict it to integers: .
- “From 2 to 7 inclusive” means .
- So the set is
Worked example
Write “natural numbers less than 5” in set-builder notation.
- Use .
- “Less than 5” means .
- So the set is
- Since this course includes 0 in , the listed form is
A good mental trick: set-builder notation is like a game filter. You are telling maths, “show me every that passes these rules.”
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