Intersection Examples in Math

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

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

Concept Recap

The intersection of sets A and B is the set of all elements that belong to both A and B simultaneously, written A \cap B.

Picture two overlapping circles in a Venn diagram—the intersection is only the overlapping region where both circles cover. For example, if set A is students who play soccer and set B is students who play piano, then A \cap B is students who do both. It is the AND gate of set theory: an element must satisfy both conditions to be included.

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: x \in A \cap B if and only if x \in A AND x \in B. Intersection corresponds exactly to logical AND.

Common stuck point: If sets share nothing, intersection is empty: \{1, 2\} \cap \{3, 4\} = \emptyset.

Sense of Study hint: Try listing out the elements of each set explicitly, then check which ones appear in both.

Worked Examples

Example 1

easy
Let A = \{1, 2, 3, 4\} and B = \{3, 4, 5, 6\}. Find A \cap B.

Solution

  1. 1
    Recall the definition: A \cap B = \{x : x \in A \text{ and } x \in B\}. An element belongs to the intersection only if it appears in both sets simultaneously.
  2. 2
    Check each element of A = \{1,2,3,4\}: is 1 \in B? No. Is 2 \in B? No. Is 3 \in B = \{3,4,5,6\}? Yes. Is 4 \in B? Yes.
  3. 3
    The elements common to both sets are 3 and 4, so A \cap B = \{3,4\}.

Answer

A \cap B = \{3, 4\}
The intersection keeps only the elements shared by both sets. It is always a subset of each original set.

Example 2

medium
Let A = \{x \in \mathbb{R} : x \ge 1\} and B = \{x \in \mathbb{R} : x \le 4\}. Find A \cap B.

Practice Problems

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

Example 1

easy
Let M = \{2, 4, 6, 8\} and N = \{1, 2, 3, 4\}. Find M \cap N.

Example 2

easy
Find A \cap B if A = \{2, 4, 6, 8\} and B = \{1, 2, 3, 4, 5\}.
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Related Concepts

SetUnionVenn Diagram

Background Knowledge

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

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