Lecture 004

Quantifiers

Existential Quantifier: $(\exists x \in S)P(x)$ means "there exists an $x$ in $S$ such that $P(x)$ holds." (某个)

• $(\exists x \in \mathbb{R})(x^2 \geq 8)$

• $\bigcup_{i\in I}A_i=\{x\in U | (\exists i \in I)(x \in A_i)\}$

• $\mathcal{E} = \{x\in \mathbb{Z}|(\exists y \in \mathbb{Z})(x=2y)\}$

• $(\exists x, y \in \mathbb{Q})(x^2+y^2=0)=(\exists x \in \mathbb{Q})((\exists y \in \mathbb{Q})(x^2+y^2=0))=(\exists y \in \mathbb{Q})(\exists x \in \mathbb{Q})(x^2+y^2=0)$

• :warning: existential quantifiers itself can switch places

Universal Quantifier: $(\forall x \in S)P(x)$ mean "for every x in S, $P(x)$ holds true". (全部)

• $(\forall x \in \mathbb{R})(x^2 \geq 8)$

• $\bigcap_{i\in I}A_i=\{x\in U | (\forall i \in I)(x\in A_i)\}$

• $(\forall p \in \mathbb{Z}^+)(\forall q \in \mathbb{Z}^+)(\sqrt{2} \neq \frac{p}{q}) = (\forall p, q \in \mathbb{Z}^+)(\sqrt{2} \neq \frac{p}{q})$

• :warning: universal quantifiers itself can switch places

Alternating Quantifiers

When both $\exists$ and $\forall$ in a statement, order matters. "All people loves some people"

Let $P(x, y)$ denotes "x loves y"

1. $(\forall x \in U)(\exists y \in U)P(x, y)$ means "everybody loves somebody."
2. $(\exists y \in U)(\forall x \in U)P(x, y)$ means "there is somebody who is loved by everybody."

Truth Tables and Logical Connectives

Truth Table: a table with one row for each possible combination of truth values for the elementary propositions (sentence symbols) and columns containing the truth values for compound statements formed from them.

Negation($\lnot$, $\neg$, $\sim$) Conjunction($\land$, $\wedge$, $\&$) Disjunction($\lor$, $\vee$, $\parallel$) Implication($\implies$, $\to$, $\Rightarrow$, $\rightarrow$, $\supset$): $P \implies Q \equiv \lnot P \lor Q$

• :warning: implies does not mean causality

• Modus Ponens: "If P->Q, and P is True, then Q must be true"

P	Q	P->Q
T	T	T
T	F	F
F	T	T
F	F	T
> When P is F, it has nothing to do with implication

• When Q is True independently, result always True

• When P is False independently, result always True

• P->Q only if P=True and Q=False

Examples:

• $A\cap B = \{x\in U | x=A \land x\in B\}$

• $(\forall x \in \mathbb{R})(x<0 \lor (\exists y \in \mathbb{R})(x=y^2))$

• $(\exists x \in \mathbb{R})(x > 2 \implies x^2 > 4)$ - True

• $(\exists x \in \mathbb{R})(x^2 > 4 \implies x > 2)$ - False

• $(\forall x \in \mathbb{N})(x \geq 0) \implies (\forall x \in \mathbb{R})(x^2 \geq 0)$ - True(no causality)

• $(\forall x \in \mathbb{Z})((\exists n \in \mathbb{Z})(2x=2n+1)\implies x^2<0)$ - True (vacuously true when P is always false) (for any integer, 2 times integer can never be an odd number)

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Homework due by noon Friday. TODO go to office hour