What is a set?
A set is a collection of things. The collection must be clear: everyone must agree if a thing is in it or not.
- Days of the week: a set (7 things, clear).
- Vowels of English: {a, e, i, o, u}. A set.
- Prime numbers below 10: {2, 3, 5, 7}. A set.
- Tasty fruits: NOT a set. People disagree about tasty.
Each thing in a set is called an element. We write sets inside curly brackets { } with commas. Order does not matter and we do not repeat things: {1, 2, 3} is the same set as {3, 2, 1}.
In step 1 of the 3D, the blue ring is the set A and the balls inside are its elements. Sets are usually named with capital letters: A, B, C.
Element belongs / does not belong to a set
To say a thing is in a set, we use the symbol ∈. It reads "belongs to" or "is an element of".
- 4 ∈ A means 4 is in A.
- 7 ∉ A means 7 is not in A. The slash cancels the ∈.
In step 2, tap the balls. A ball inside the ring shows ∈. A ball outside the ring shows ∉. Always check the whole list of A before you decide.
Counting: n(A) means the number of elements in A. For A = {2, 4, 6, 8, 10}, n(A) = 5.
Subsets
Set B is a subset of set A when every element of B is also an element of A. We write B ⊂ A.
How to test
- Take the elements of B one by one.
- Look for each one in A.
- If all are found, B ⊂ A. If one is missing, B ⊄ A.
Step 3 shows B = {2, 4}: both are in A, so B ⊂ A. Step 4 shows C = {4, 5}: 5 is not in A, so C ⊄ A.
Two special subsets
- Every set is a subset of itself: A ⊂ A.
- The empty set ∅ = { } has no elements, so nothing can be outside. It is a subset of every set.
If B ⊂ A and B is not equal to A, then B is a proper subset. Subsets can be nested, like in step 5: {4, 8} inside the even numbers inside {1, …, 10}.
More examples of sets and subsets
- Months with 31 days ⊂ all months of the year.
- Natural numbers {1, 2, 3, …} ⊂ whole numbers {0, 1, 2, …}.
- Colours of the Indian flag {saffron, white, green, blue} is a set of 4 colours; {saffron, green} is a subset.
- A = {x : x is a letter in "MATHS"} = {M, A, T, H, S}.
The number of subsets of a set with n elements is 2ⁿ. For {a, b}: ∅, {a}, {b}, {a, b} = 4 subsets = 2².
Try it
At home: put 6 spoons, forks and cups on a table: this is your set. Draw a ring with a string around only the spoons. Is the set of spoons a subset of all things? Now pick one spoon and one cup: is that pair a subset of the spoons?
In the 3D: before you tap, guess if your set is a subset of A. Then tap and check. Can you build a set with 3 balls that is a subset? One that is not?
Key formulas and definitions
- x ∈ A: x is an element of A
- x ∉ A: x is not in A
- B ⊂ A: every element of B is in A
- n(A): number of elements of A
- A set with n elements has 2ⁿ subsets
Worked examples
1. A = {1, 2, 3, 4, 5}. Is 3 ∈ A? Is 6 ∈ A?
3 is in the list, so 3 ∈ A. 6 is not in the list, so 6 ∉ A.
2. Is {1, 3} a subset of A = {1, 2, 3, 4, 5}?
Take 1: it is in A. Take 3: it is in A. Nobody is outside, so {1, 3} ⊂ A.
3. Is {2, 6} a subset of A = {1, 2, 3, 4, 5}?
2 is in A, but 6 is not. One element is outside, so {2, 6} ⊄ A.
4. Write all subsets of {x, y}.
Empty set ∅, then {x}, then {y}, then {x, y}. That is 4 subsets, and 2² = 4. Check.
Common mistakes
- Using ∈ between two sets. 4 ∈ A is for one element; {4} ⊂ A is for a set. Element goes with ∈, set goes with ⊂.
- Thinking the empty set is not a subset. It is a subset of every set, because no element is outside.
- Forgetting that A ⊂ A is true. Every set is a subset of itself.
- Repeating elements, like {1, 1, 2}. Write each element once: {1, 2}.