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Proof

(related to Proposition: Intersection of a Set With Another Set is Subset of This Set)

Context

• Let $A$ and $B$ be sets.
• Let $A\cap B$ be the intersection of $A$ and $B$.
• We will prove $A\cap B\subseteq A$. The proof for $A\cap B\subseteq B$ is similar.

Hypothesis

• Let $x\in A\cap B$.

Implications

• By the definition of intersection, it is true that $x\in A\wedge x\in B.$
• By the truth table of conjunction "and" ("$\wedge$") it must be true that $x\in A$.

Conclusion

• Since $x\in A$ for all $x\in A\cap B$, then by definition of subset $A\cap B\subseteq A$.
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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016