Quantified Statements

MATH1061

Predicates & Valid Arguments

Predicates are able to be used in different scenarios to determine quantified statements and common arguments.

Activity 1:

1. $q⟹p$
2. $\sim p⟷r$
3. $\sim s⟹q$ (or $\sim q⟹s$)
4. $\sim s$
5. $\therefore \sim r$

Solution 1: Check whether the argument is invalid, by assuming conclusion is false and all premises are true.

$$\begin{array}{c}\begin{array}{r}\text{Suppose all the premises are true and the conclusion is false.}\\ \text{Since the conclusion}\sim r\text{is false,}r\text{is true.}\\ \text{Since 4 is true,}\sim s\text{is true and so}s\text{is false.}\end{array}\end{array}$$ $$\begin{array}{c}\begin{array}{r}\begin{array}{c}\{\begin{array}{ll}1.& q⟹p\\ 2.& \sim p⟷r\\ 3.& \sim s⟹q\\ 4.& \sim s\end{array}\\ \\ \text{Goal: Get the conclusion}\sim r\\ \\ \{\begin{array}{llc}5.& q& \text{from 3 and 4, using Modus Ponens}\\ 6.& p& \text{from 1 and 5 using Moddus Ponens}\\ 7.& \sim r& \text{from 6 and 2 by definition of bidirectional arrow}⟷\\ 8.& r⟹\sim p& \text{from 7 by specialisation}\\ 9.& \sim r& \text{from 8 and 6 by Modus Tollens (Double Negative Law)}\end{array}\end{array}\end{array}\end{array}$$

Quantified Statements

The symbol $\forall$ means “for all”.

The symbol $\exists$ means “there exists.”

Pre-class Questions

True or false? $\exists y\in \mathbb{Z}$ such that $y^2-y=0$. This statement is true as you only need one set of $f(y)$ to make the statement true. For example, $y=0$ or $y=1$ makes the statement true.

True or false? $\forall x\in \mathbb{Z}$, if 6 is divisible by $x$, then $x=2$. This statement is false as you need all sets of numbers in the range of the integer to make the statement true. In this case, consider $x=3$. We have $3\in \mathbb{Z}$, and 6 is divisible by 3, but $3\ne 2$.