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Prove that for any sets A and B, A is a subset of A union B.

April 19, 2026Mathematics

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Answer

\text{Thus, } A \subseteq A \cup B \text{ is proven.}

4 steps

State the property to be proven.
We want to prove that for any set $A$ and set $B$ , $A$ is a subset of $A \cup B$ . This is written as $A \subseteq A \cup B$ .
Assume an arbitrary element is in set $A$ .
Let $x$ be an arbitrary element such that $x \in A$ .
Apply the definition of set union.
The definition of set union states that an element belongs to $A \cup B$ if it belongs to $A$ or it belongs to $B$ .
Since we assumed $x \in A$ , the condition " $x \in A$ or $x \in B$ " is true.
Therefore, $x \in A \cup B$ .
Conclude the subset relationship.
Since every element $x$ that belongs to $A$ also belongs to $A \cup B$ , by the definition of a subset, we can conclude that $A \subseteq A \cup B$ .
$\text{Thus, } A \subseteq A \cup B \text{ is proven.}$

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