Element Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Element.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An individual object that belongs to, or is a member of, a given set — either it is in the set or it is not.

An element is simply one item inside the collection — either it is in, or it is out. There is no "partially in."

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Membership is a yes-or-no relation: either x \in A or x \notin A, never both.

Common stuck point: \{1\} \in \{\{1\}, 2, 3\} but 1 \notin \{\{1\}, 2, 3\}. The set \{1\} is different from the element 1.

Sense of Study hint: Ask yourself: 'Am I asking if this OBJECT is in the set, or if this SET is contained in the set?' That tells you whether to use the element-of or subset symbol.

Worked Examples

Example 1

easy
Let A = \{3, 7, 11, 15\}. Determine whether 7 \in A, \{7\} \in A, and 10 \notin A.

Solution

  1. 1
    Check 7 \in A: the element 7 appears in the listing \{3, 7, 11, 15\}, so 7 \in A. True.
  2. 2
    Check \{7\} \in A: the object \{7\} is a set, not a number. The set A does not contain \{7\} as a member, only the number 7. So \{7\} \notin A.
  3. 3
    Check 10 \notin A: 10 does not appear in the listing, so indeed 10 \notin A. True.

Answer

7 \in A,\quad \{7\} \notin A,\quad 10 \notin A
The symbol \in tests whether an object is a direct member of a set. A set \{7\} and the number 7 are different objects — confusing them is the most common mistake with element notation.

Example 2

medium
Let B = \{\emptyset, \{1\}, \{2, 3\}\}. Which of the following are true? (a) \emptyset \in B, (b) 1 \in B, (c) \{1\} \in B, (d) \{2,3\} \subseteq B.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Let S = \{1, 2, 3, 4, 5\}. Fill in each blank with \in or \notin: (a) 3 \;\square\; S, (b) 6 \;\square\; S, (c) 0 \;\square\; S.

Example 2

medium
Let T = \{x \in \mathbb{Z} : x^2 < 10\}. List all elements of T and state |T|.
← Back to Element Formula Explained Practice Mode

Related Concepts

SetSubset

Background Knowledge

These ideas may be useful before you work through the harder examples.

set