Subset Examples in Math

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

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

Concept Recap

Set A is a subset of set B if every element of A is also an element of B, written A \subseteq B.

Every single thing in A can also be found inside B. Think of A as fitting entirely within B, like a small circle inside a big one.

Core idea: A \subseteq B means: for every x, if x \in A then x \in B. Equivalently, A \cap B = A.

Common stuck point: \emptyset (empty set) is a subset of every set. Every set is a subset of itself.

Sense of Study hint: Try picking each element of A one at a time and checking if it appears in B. If every single one passes, A is a subset of B.

easy

Let A = \{1, 2, 3, 4, 5\} and B = \{2, 4\}. Determine whether B \subseteq A.

1
Recall the definition: B \subseteq A means every element of B is also an element of A. We check each element of B individually.
2
Check 2 \in B: is 2 \in A = \{1,2,3,4,5\}? Yes. Check 4 \in B: is 4 \in A? Yes.
3
Since every element of B belongs to A, we conclude B \subseteq A. Note also B \ne A since A has elements (1, 3, 5) not in B, so B is a proper subset: B \subsetneq A.

Answer

B \subseteq A

B is a subset of A if every element of B belongs to A. We verify this by checking membership one element at a time.

medium

List all subsets of S = \{a, b, c\}.

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

easy

Let X = \{1, 2, 3\} and Y = \{1, 2, 3, 4, 5\}. Is X \subseteq Y? Is Y \subseteq X?

easy

Let A = \{1, 3\} and B = \{1, 2, 3, 4\}. Is A \subseteq B? Is B \subseteq A?

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

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