Number Sets (ℕ, ℤ, ℚ, ℝ): Union & Intersection

Master union and intersection of sets with step-by-step guidance and interactive exercises.

Union & Intersection

Solved: 0
Find the resulting set of the union (∪) or intersection (∩) of the two sets below. Write the numbers inside curly braces separated by commas, e.g. {1, 2, 3}, or type ∅ for the empty set.
Learning Topics

Set Operations: Union (∪) & Intersection (∩)

We can combine two sets using two key operations:

• ∪ (Union) — "everything together": The union A ∪ B contains all elements that appear in A, in B, or in both. Think of it as merging two groups.
  Example: {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5} (no duplicates!)

• ∩ (Intersection) — "what they share": The intersection A ∩ B contains only the elements that appear in both A and B. Think of it as the overlap.
  Example: {1, 2, 3} ∩ {3, 4, 5} = {3} (only 3 is in both)

• ∅ (Empty Set): When two sets share nothing, their intersection is the empty set ∅.
  Example: {1, 2} ∩ {3, 4} = ∅

Tip: Union = more (or equal) elements. Intersection = fewer (or equal) elements.
Learning Topics

Frequently Asked Questions

What is a set and what are its elements?

A set is a collection of distinct objects or numbers called elements. We write a set by listing its elements inside curly braces, like A = {1, 2, 3}.

What do the symbols ∈ and ∉ mean?

The symbol ∈ means “belongs to” or “is an element of” (e.g. 2 ∈ {1, 2, 3} is true). The symbol ∉ means “does not belong to” (e.g. 4 ∉ {1, 2, 3} is true).

What is the union of two sets?

The union A ∪ B contains all elements that appear in A, in B, or in both. No duplicates are listed. Example: {1, 2} ∪ {2, 3} = {1, 2, 3}.

What is the intersection of two sets?

The intersection A ∩ B contains only elements that appear in both A and B. Example: {1, 2, 3} ∩ {2, 3, 4} = {2, 3}.

What are Integers (ℤ)?

Integers include all whole numbers — positive, negative, and zero: ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}. Every natural number is also an integer, but not every integer is natural.

What is the difference between real and complex numbers?

Real numbers include all rational and irrational numbers. Complex numbers include all real numbers as well as numbers containing the imaginary unit $i$ (where $i^2 = -1$), allowing us to solve roots of negative numbers.

Is zero positive or negative?

Zero is neither positive nor negative. It is the boundary between them.