To Prove: | Do: | ||
$P\implies Q$ | "Assume $P$ is true," prove $Q$ is true, or "Assume $Q$ is false," prove $P$ is false, or "Assume $P$ is true and $Q$ is false", produce a contradiction. | ||
$P\iff Q$ |
Prove $(P\implies Q)\And(Q\implies P)$, or prove $(P\implies Q)\And(\Not P\implies\Not Q)$, or prove $(\Not Q\implies\Not P)\And(Q\implies P)$, or prove $(\Not Q\implies\Not P)\And(\Not P\implies\Not Q)$ | ||
$(\forall x)(P(x))$ | "Let $x$ be an arbitrary $\ldots$" Prove $P(x)$. | ||
$(\exists x)(P(x))$ | "Take $x = \ldots$" Prove $P(x)$ for this $x$. |
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$A \subseteq B$ | Prove $(\forall x\in A)(x\in B)$ i.e., if $x\in A$ then $x\in B$. |
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$A = B$ | Prove $(A \subseteq B) \And (B \subseteq A)$. | ||
$A = \emptyset$ | Prove $(\forall x)(x\notin A)$ (frequently best to use proof by contradiction). |
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$x\in A \cup B$ | Prove $(x\in A) \Or (x\in B)$. | ||
$x\in A \cap B$ | Prove $(x\in A) \And (x\in B)$. | ||
$x\in A - B$ | Prove $(x\in A) \And (x\notin B)$. | ||
$\Not(P(x) \implies Q(x))$ | Prove $(\exists x)(P(x)\And\Not Q(x))$. | ||
$\Not (P(x) \iff Q(x))$ | Prove $(\exists x)(P(x)\And\Not Q(x))\Or(\exists x)(Q(x)\And\Not P(x))$. | ||
$\Not(\exists x)(P(x))$ | Prove $(\forall x)(\Not P(x))$. | ||
$\Not(\forall x)(P(x))$ | Prove $(\exists x)(\Not P(x))$. | ||
$A\not\subseteq B$ | Prove $(\exists x)(x\in A \And x\notin B)$. | ||
$A \ne B$ | Prove $(A\not\subseteq B) \Or (B\not\subseteq A)$. ie, there is an $x\in A$ where $x\notin B$ or there is an $x\in B$ where $x\notin A$. |
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$A \ne \emptyset$ | Prove $(\exists x)(x\in A)$. | ||
$x\notin A \cup B$ | Prove $(x\notin A) \And (x\notin B)$. | ||
$x\notin A \cap B$ | Prove $(x\notin A) \Or (x\notin B)$. | ||
$x\notin A - B$ | Prove $(x\notin A) \Or (x\in B)$. | ||
To prove $P$ by contradiction:
"Assume $P$ is false"
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