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I'm currently solving a quite specific problem and in the final step I made a statement that can be generalised such that: $$((|A|=|B|)\wedge(A\subset B)) \implies (A=B)$$ Whilst this is clearly intuitively true for finite sets $A$ and $B$ and probably a safe statement for my purposes I was wondering if:

  1. Is it mathematically sound to make such a statement for finite sets or should there be some further proof?
  2. Is it the statement mathematically true in general including infinite sets?
Martin Sleziak
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J-S
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1 Answers1

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  1. Hint: Use induction, and you can get a formal proof of this statement: Assume that $|A| = |B|=n$ and $A \subset B$. Since $|A| = |B|$, we note that there exists a bijection $f:A \to B$. Assume that $A \neq B$. Since $A \subset B \implies$ $\exists b \in B$ such that $b \not \in A$. But then, since $B$ is finite, we can show that $|A| < |B|$ (through induction) $\implies$ contradiction.

  2. This is not true for infinite sets. Consider $B=\mathbb{N}$ and $A=\mathbb{N} - \{1\}$. Clearly, $A \subset B$, and one can show pretty easily that $|A| = |B|$. However, $A \neq B$.

Kyle Gannon
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