Introduction to statements and sets

Set#

A set is a well-defined, unordered collection of distinct objects.

Each object that appears in this collection is called an element (or member) of the set.

Sets can contain any type of object.

Some examples of sets#

$$\begin{aligned} \mathbb{R} &\text{: real numbers} \ \natnums &\text{: Natural Numbers, whole numbers from }0 \ \mathbb{Z} &\text{: Integers} \ \mathbb{Q} &\text{: Rational Numbers}\end{aligned}$$

We use capital letter sot represent a set, $S$, and small letters to represent elements of a set, $s$.

$\in$ to denote membership, $\notin$ to denote non-membership.

Statements#

A statement is a sentence that has a definite state of being either true or false.

Examples of Statements#

$$\begin{aligned}2 + 3 &= 5 \text{ False} \ \pi + 2 &\leq 5 \text{ True}\end{aligned}$$

We can negate statements, $\neg{A}$.

Double negation brings us back to the start. Or in Math speak, logically equivalent.

$$\neg(\neg{A}) \equiv A \text{ (Note the tripe equal sign)}$$

Quantified statements#

A quantified statement contains four parts:

• A quantifier (universal $\forall$, or existential $\exists$)
• A variable (any symbol representing a quantity or mathematical object),
• A domain (any set)
• An open sentence involving the variable (that is either true or false whenever a value of the variable chosen from the domain is specified).

One example is:

$$\forall{n} \geq 5, n^2 + 2n \leq 20n$$

The first part of the sentence contains the quantifier $\forall$, the variable $n$, and the domain $s \in {5, 6, ,7, \ldots}$.

The second part of the statement contains the open sentence $n^2 + 2n \leq 20n$. An open sentence simply means that the statement depends on some variables and the truth of the statement cannot be determined.

Describing statements#

Universal Quantified statements: $$\forall x \in S, P(x)$$ In English: For all $x$ in $S$, $P(x)$ is true. False otherwise.

Existential Quantified statements: $$\exist x \in S, P(x)$$ In English: There exist at least one value of $x$ in $S$ such that $P(x)$ is true.

Negating Statements#

Negating Universal Quantified Statements $$\neg (\forall x\in S, P(x)) \equiv \exist x \in S, \neg (P(x))$$

Negating Existential Quantified Statements $$\neg (\exist x\in S, P(x)) \equiv \forall x \in S, \neg (P(x))$$

Multiple quantifiers#

Quantifiers in a statement containing more than one quantifier are called nested quantifiers.

The order in which they appear will change the meaning of these statements.

For all first, then exist#

$$\forall x \in X, \exist y \in Y, x > y$$ The quantified statement above is true since we are choosing a single y value after the set $X$ has been establish.

Therefore as long as $y$ is $\min{(X)} - 1$, the above statement will always be true.

Exist first, then for all#

$$\exist x \in X, \forall y \in Y, x > y$$ The above statement is now false because regardless off what $x$ value we choose, there is always a value that $y$ can choose such that $y>x$.

Going one step further $$\forall x \in X, \exist y \in Y, x > y \not\equiv \exist x \in X, \forall y \in Y, x > y$$

The open statement $x > y$ can also be replaced with $Q(x, y)$ to mean an open set whose truth value can be determine for $x$ and $y$, chosen from the domain $X$, and $Y$ respectively

More than 2 quantifiers#

Consider:

$$\forall x \in X, \exist y \in Y, \forall z \in Z, R(x, y, z)$$

This statement can be rewritten as:

$$\left.\begin{aligned} &\forall x \in X, P(X) \ \text{where }P(X) \text{ is } &\exist y \in Y, Q(x,y) \ \text{where } Q(x,y) \text{ is } &\forall z \in Z, R(x,y,z)\end{aligned}\right\rbrace$$

Notice the nesting.

Such a scenario is especially common in deriving limits for calculus.

Limits in calculus

The limit, $L$, in calculus is defined such that as $x$ approaches $a\in \mathbb{R}$, $f(x)$ tends towards $L$.

More formally, for any positive tolerance, $\epsilon$, from $L$, there exist a $\delta \gt 0$, such that if $|x - a|\lt \delta$ is true, then $|f(x) - L| \lt \epsilon$.

Even more formally, $\forall \epsilon \in \mathbb{R}, \exist \delta \in \mathbb{R}, \forall x \in \mathbb{R}, if |x - a|\lt \delta, then |f(x) -L| < \epsilon$

Negating Multiple Quantifiers#

Negating quantifiers is simply a matter of working from right to left and doing one of the following:

• Changing $\forall$ into $\exist$ or vice versa
• Changing the open statement to the opposite, $= \text{ becomes } \neq, \lt \text{ becomes } \geq$