Number Sets (ℕ, ℤ, ℚ, ℝ): Belonging to Sets

Learn what it means for an element to belong to a set. Practice the ∈ and ∉ symbols with concrete, visual exercises.

Belonging to Sets

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A set A is shown below. Does the highlighted element belong to A (∈) or not (∉)?
Learning Topics

What is a Set? Belonging (∈) and Not Belonging (∉)

A set is simply a collection of distinct objects — called elements or members. We write a set by listing its elements inside curly braces, like A = {2, 5, 8, 11}.

We use two special symbols to describe membership:
• ∈ (belongs to / is an element of): We write 5 ∈ A to say "5 is in set A". Since 5 is listed inside A above, this is true.
• ∉ (does not belong to): We write 7 ∉ A to say "7 is not in set A". Since 7 is not listed, this is true.

Key idea: To check membership, just look at the list! If the element appears inside the curly braces, it belongs (∈). If it doesn't appear, it does not belong (∉).

Example: For B = {1, 3, 5, 7, 9}:
• 3 ∈ B ✓ (3 is in the list)
• 4 ∉ B ✓ (4 is not in the list)
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.