Math 199 (Notes)

 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 $x$. $A \subseteq B$ Prove $(\forall x\in A)(x\in B)$ i.e., if $x\in A$ then $x\in B$. $A = B$ Prove $(A \subseteq B) \And (B \subseteq A)$. $A = \emptyset$ Prove $(\forall x)(x\notin A)$ (frequently best to use proof by contradiction). $x\in A \cup B$ 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 $x\notin A$. $A \ne \emptyset$ 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" then show that you arrive at a contradition. To prove $(\forall n\in\N)(P(n))$ by Mathematical Induction, show the following: $P(1)$ is true For all $k\in\N$, if $P(k)$ is true then $P(k+1)$ is true.
 Math 199 (Fall 2005) web pages Created: 01 Sep 2005 Last modified: 11 Oct 2005 10:31:53 Comments to: dpvc@union.edu