L2 · Set language: elements, subsets, set-builder notation

Learning goal

“I can use the symbols ∈, ∉, ⊆ and ∅ correctly, list the elements of a set written in set-builder notation, and write a described set in set-builder notation.”

Element — one object in a set. Membership is written with ∈: 4 ∈ A means "4 is an element of A"; 5 ∉ A means "5 is not an element of A."

Subset — set A is a subset of set B, written A ⊆ B, when EVERY element of A is also an element of B.

Empty set — the set with no elements, written ∅ or { }. It is not the same as {0}, which has one element.

Set-builder notation — a rule for building a set: {x | condition on x} reads "the set of all x such that the condition holds."

Worked example

Let A = {2, 4, 6, 8}. Step 1: Membership: 4 ∈ A (4 is listed); 5 ∉ A (5 is not listed). Step 2: A is a finite set - it has exactly 4 elements. The set of all even natural numbers is infinite - the listing never ends: {2, 4, 6, 8, ...}. Step 3: Subset test: is {2, 6} ⊆ A? Check EVERY element: 2 ∈ A and 6 ∈ A, so yes. Is {2, 10} ⊆ A? 10 ∉ A, so no - one failing element is enough to fail the test. Step 4: Equality: {2, 4, 6} and {6, 2, 4} are the SAME set - order does not matter, and each element is listed once even if a roster writes it twice.

Check yourself — explain it

Why is {0} not the empty set?

Paired problem — example still visible

For B = {1, 3, 5, 7, 9}: is 7 ∈ B? Is B finite or infinite? -> 7 ∈ B; B is finite (5 elements).

Faded completion — last step is yours

"The set of all even natural numbers is ______ (finite / infinite), because its listing ______." -> infinite; never ends

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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