What is a set?
A set is a well-defined collection of distinct objects. Each object in a set is called an element (or member). "Well-defined" means we can always tell clearly whether something belongs to the set or not.
We name sets with capital letters and list their elements inside curly brackets { }, separated by commas. For example, A = {2, 4, 6, 8}. Elements are not repeated and the order does not matter.
Key idea / 要点
∈ means "is an element of" and ∉ means "is not an element of". The number of elements in set A is written n(A). The empty set has no elements and is written ∅ or { }.
Worked example
Worked example / 例题
Let A be the set of even numbers less than 10.
By listing: A = {2, 4, 6, 8}.
So n(A) = 4. Here 6 ∈ A (6 is an element), but 5 ∉ A (5 is not an element).
Describing a set
You can describe a set in words ("the vowels in English"), by listing ({a, e, i, o, u}), or by a rule. Two sets are equal if they have exactly the same elements, so {1, 2, 3} = {3, 2, 1}. Because order and repetition do not matter, {1, 2, 2, 3} is really the same set as {1, 2, 3}; we simply write each element once. When you count elements for n(A), count each distinct object only once, and remember that the empty set has n(∅) = 0.
Remember / 记住
- Use { } for a set and capital letters for its name.
- No repeats; order does not matter.
- n(A) counts the elements; ∅ is the empty set.