Real Analysis

Mathematical logic

Proposition

A statement in either true or false state.

Symbols

Symbol	Read as
$\land$	and
$\lor$	or
$\rightarrow$	then
$\implies$	implies
$\Leftarrow$	implied by
$\iff$	if and only if
$\forall$	for all
$\exists$	there exists
$\sim$	not

Let’s take $a \to b$. $$

1. Contrapositive or transposition: $\sim b \to\,\, \sim a$. This is equivalent to the original.
2. Inverse: $\sim a \to\,\,\sim b$. Does not depend on the original.
3. Converse: $b \to a$. Does not depend on the original.

$a \to b \,\equiv\,\,\sim a \lor b \,\equiv\,\, \sim b \to\,\,\sim a$

Required proofs

• $\sim\forall x \, P(x) \equiv \exists x \sim{P(x)}$
• $\sim\exists x \, P(x) \equiv \forall x \sim{P(x)}$
• $\exists x\, \exists y P(x,y) \equiv \exists y\, \exists x {P(x,y)}$
• $\forall x\, \forall y P(x,y) \equiv \forall y\, \forall x {P(x,y)}$
• $\exists x\, \forall y P(x,y) \implies \forall y\, \exists x {P(x,y)}$
• $(A \rightarrow C)\land(B \rightarrow C) \equiv (A\lor B)\rightarrow C$

Methods of proofs

1. Just proof what should be proven
2. Prove the contrapositive
3. Proof by contradiction
4. Proof by induction

Proof by contradiction

Suppose $a \implies b$ has to be proven. If $a\,\land \sim b$ is proven to be false, then, by proof by contradiction, $a \implies b$ can be trivially proven.

Logic behind proof by contradiction

$a\,\land \sim b = F$

$\sim{(a\,\land \sim b)} =\,\, \sim F$

$\sim a \lor b = T$

$a \to b = T$

$a \implies b$