Mathematics from zero
Sets
A fruit bowl holds an apple, a pear, and a banana. It does not matter what order they sit in, and a second apple would not make it “more of a bowl” — it is just a collection of distinct things. Mathematics calls such a collection a set.
After this lesson you can say what a set is, decide whether something is in a set, give the size of a set, and combine two sets by union and intersection.
A set is a collection of distinct things. The things in a set are its
elements. We write a set by listing its elements inside curly brackets: the set
{1, 2, 3} has the elements 1, 2, and 3. Order does not matter, and an element is
either in or out — listing it twice changes nothing.
The basic question about a set is membership: is this thing in it? For the set
{1, 2, 3}, the number 2 is an element — it is in the set. The number 5 is not an
element — it is out. Every set splits the world into two: things that belong, and
things that do not.
The size of a set is how many elements it has. The set {1, 2, 3} has size 3. A
set can also be empty — the empty set has no elements at all, and its size is 0.
The empty set is still a perfectly good set; it is just the collection of nothing.
Two sets combine by union and intersection. The union of two sets is everything that is in either one — all the elements gathered together, each listed once. The intersection is everything that is in both — only the elements the two sets share. Union is the generous combination; intersection is the strict one.
Set A is {1, 2, 3} and set B is {2, 3, 4}. Find their union and their
intersection.
The union gathers every element that is in A or in B, each counted once. From A:
1, 2, 3. From B the new ones: 4. The union is {1, 2, 3, 4} — size 4.
The intersection keeps only elements in both A and B. Is 1 in both? No, only A.
Is 2 in both? Yes. Is 3 in both? Yes. Is 4 in both? No, only B. The intersection is
{2, 3} — size 2.
Union is always at least as big as either set; intersection is always at most as big as the smaller set.
Why this works
Why does listing an element twice change nothing? Because a set only records whether
something belongs, not how many times. Membership is a yes-or-no question — like the
truth values from earlier in this unit. {1, 1, 2} and {1, 2} are the same set:
both answer “is 1 in?” with yes and “is 2 in?” with yes. There is no “how many copies”
to track.
Common mistake
A common mistake is mixing up union and intersection — taking the intersection of
{1,2,3} and {2,3,4} to be the larger {1,2,3,4}. Intersection keeps only the
shared elements, so it is the smaller {2,3}. Union gathers everything; intersection
keeps only the overlap. When in doubt: union means “or”, intersection means “and”.
How many elements are in the set {2, 4, 6, 8}? Type the size.
What is the size of the empty set? Type the number.
A is {1, 2, 3} and B is {3, 4}. What is the size of their union? Type the number.
A is {1, 2, 3} and B is {3, 4}. What is the size of their intersection? Type the number.
Is the number 5 an element of the set {1, 2, 3}? Type 1 for yes, 0 for no.
What is the intersection of two sets?
A set is a collection of distinct things called its elements, written inside curly brackets; order and repeats do not matter. The basic question about a set is membership — whether a thing is in it. The size of a set is its element count, and the empty set has size 0. Two sets combine by union (everything in either) and intersection (only what is in both).