Set language · Definition

Set

setsfoundations

A set is the basic way to group objects in mathematics. Here you will see what it means to belong to a set and when two sets are equal.

Definition

A set is a well-defined collection of objects, called elements. Whether an object belongs to the set must be decidable without ambiguity.

\[x \in A \lor x \notin A\]

Also called: well-defined collection

A set is determined only by its elements: two sets are equal when they have exactly the same elements, regardless of order or repetition. For example, \(\{1,2,2\} = \{2,1\}\). Also, the empty set \(\varnothing\) has no elements and is contained in every set.

Take \(A = \{a, b, c\}\). Then \(b \in A\) because b is an element of A, while \(d \notin A\) because d does not appear in the list. Elements are counted without repetition: \(A\) has three elements.

A typical mistake is confusing membership with containment: writing \(a \subseteq A\) when one means \(a \in A\). If \(a\) is an element, it is not a set and cannot be contained in A; \(\{a\} \subseteq A\) is correct.

Think about it

is \(\{1,2\}\) equal to \(\{2,1\}\)?

Yes, because both have exactly the same elements; order does not matter.

Exercises 10

  1. Which of the following best describes a set?
  2. If x ∈ A, what does it mean?
  3. If A = {1, 2, 3}, which statement is true?
  4. Which symbol represents the empty set?
  5. How many elements does A = {a, b} have?
  6. If \(A = \{1,2,3\}\), which statement is true?
  7. Two sets are equal when...
  8. Which statement is true?
  9. If A = {∅}, then...
  10. Which correctly distinguishes membership from containment?

References

  • I. N. Herstein, Topics in Algebra, 2.ª edición, John Wiley & Sons, 1975.