Sets and Real Numbers – Lesson 1

A comprehensive set of vocabulary flashcards covering fundamental terminology and notation for sets, subsets, operations on sets, and classifications within the real number system.

Set

A well-defined collection of distinct objects treated as a single entity.

Roster Method

Listing every element of a set inside braces, e.g., {1,2,3}.

Rule / Set-Builder Method

Describing a set by stating a property its elements satisfy, e.g., {x | x is even}.

Element (∈)

A member of a set; a ∈ A means a is in set A.

Empty (Null) Set Ø

The unique set containing no elements, {}.

Finite Set

A set with a limited, countable number of elements.

Infinite Set

A set with unlimited, uncountable, or endlessly countable elements.

Universal Set (U)

The set containing all objects under discussion.

Equal Sets

Two sets containing exactly the same elements.

Equivalent Sets

Sets having the same number of elements, regardless of what they are.

Joint Sets

Sets that share at least one common element.

Disjoint Sets

Sets that share no common elements.

Proper Subset (⊂)

Set A is a proper subset of B if every element of A is in B and A ≠ B.

Improper Subset

A subset that is either the entire set itself or the empty set.

Superset (⊃)

Set B is a superset of A if A ⊂ B.

Union of Sets (A ∪ B)

All elements in A or B (or both).

Intersection (A ∩ B)

All elements common to both A and B.

Complement (A′)

Elements in the universal set U that are not in A.

Difference (A − B)

Elements in A that are not in B.

Natural Numbers (ℕ)

Counting numbers {1,2,3,…}; sometimes 0 is included.

Whole Numbers

Natural numbers together with 0: {0,1,2,3,…}.

Integers (ℤ)

Positive and negative whole numbers and 0: {…,−2,−1,0,1,2,…}.

Positive Integers (ℤ⁺)

Integers greater than 0: {1,2,3,…}.

Negative Integers (ℤ⁻)

Integers less than 0: {…,-3,-2,-1}.

Rational Numbers (ℚ)

Numbers expressible as a fraction n/d with integers n, d and d ≠ 0.

Irrational Numbers

Non-terminating, non-repeating decimals not expressible as n/d, e.g., π, √2.

Real Numbers (ℝ)

All rational and irrational numbers; points on the number line.

Positive Real Numbers (ℝ⁺)

All real numbers greater than 0.

Negative Real Numbers (ℝ⁻)

All real numbers less than 0.

Complex Numbers

Numbers of the form a + bi where a, b ∈ ℝ and i² = −1.

Additive Identity

The number 0; a + 0 = a for any real number a.

Multiplicative Identity

The number 1; a · 1 = a for any real number a.

Additive Inverse

For a real number a, the number −a satisfying a + (−a) = 0.

Commutative Property

Order doesn’t affect sum or product: a+b = b+a, ab = ba.

Associative Property

Grouping doesn’t affect sum or product: a+(b+c) = (a+b)+c.

Number Line

A horizontal line representing real numbers in order, increasing to the right.

Set Notation Braces { }

Symbols used to enclose the elements of a set.

Membership Symbol (∉)

a ∉ A means a is not an element of set A.

Cardinality

The number of elements in a set.

Universal Quantifier "…"

Ellipsis indicating continuation to infinity, e.g., {1,2,3,…}.