Definition: A statement is a sentence which is decidably true or false.
Examples of Statements
The sky is blue.
The sky is yellow.
All cows eat grass.
Mr. Holt is wearing a knitted sweater today.
Examples of Non-Statements
"The Big Lebowski" is a great film.
Green is the best color.
Just take it easy, man.
Is Linda Lou going to win the student body presidency?
Since there is no objective way to determine the truth value of any of these sentences, they cannot be called statements.
Simple Versus Compound Statements
If a statement can be broken up into two or more separate statements, each with their own truth value, then the statement is called a compound statement.
Example: The statement Billy Bob is going to the party or the moonshiners convention. really contains two statements:
Billy Bob is going to the party.
Billy Bob is going to the moonshiners convention.
Simple Versus Compound Statements
If a statement isn't compound, then it is called a simple statement.
This essentially means that the statement has only one idea or thought. It can't be broken up into multiple thoughts or ideas.
Forming Compound Statements
We can create compound statements by joining statements with connectives.
The connectives we will consider in this class are:
and
or
implies
if and only if
Examples of Compound Statements: Consider the two simple statements: I will paint my bedroom blue. and I will paint my kitchen green.
Conjunction (and): I will paint my bedroom blue, and I will paint my kitchen green.
Disjunction (or): I will paint my bedroom blue, or I will paint my kitchen green.
Conditional (implies): If I paint my bedroom blue, then I will paint my kitchen green.
Biconditional (if and only if): I will paint my bedroom blue if and only if I paint my kitchen green.
Negation
Consider the statement: The sky is blue.
What statement would always have the opposite truth value?
Negation
Definition: The negation of a statement is the statement which always has the opposite truth value.
Examples
| Statement | Negation |
| Linda Lou won the student body presidency. | Linda Lou didn't win the student body presidency. |
| Billy Bob doesn't makes moonshine. | Billy Bob makes moonshine. |
Quantified Statements
Some statements contain words like "every, for all, some, there exists, there is no..., no," etcetera. These are called quantified statements.
Example: All cows eat grass.
Example: Some animals are self-aware.
Example: No politician is allowed to take bribes.
Example: There exists a politician who is allowed to take bribes.
Quantifiers
Words in a statement such as every, for all, some, there exists, there is no..., no, etcetera, are called quantifiers.
"All-or-nothing" quantifiers (every, for all, no, there is no) are called universal quantifiers.
"Existence" quantifiers (there is, there exists, some, at least one) are called existential quantifiers.
Negation of Quantified Statements
Consider the statement: All cows eat grass.
What statement would always have the opposite truth value?
Negation of Quantified Statements
Definition: the negation of a statement is the statement which always has the opposite truth value.
| Statement | Possible Negation |
| All cows eat grass. | 1) Not all cows eat grass. 2) Some cows don't eat grass. 3) There is a cow which does not eat grass. |
| Some animals are self-aware. | 1) No animals are self aware. 2) All animals are not self-aware. |
| No politician is allowed to take bribes. | 1) There exists a politician who is allowed to take bribes. 2) Some politicians are allowed to take bribes. |
A Common Error in Negating Quantified Statements
It is incorrect negate only the statement after the quantifier.
| Statement | Incorrect Negation |
| All cows eat grass. | 1) All cows don't eat grass. 2) No cows eat grass. |
| Some animals are self-aware. | Some animals are not self aware. |
| No politician is allowed to take bribes. | No politician is not allowed to take bribes. |
Using Symbols to Represent Statements
Fact: Sometimes writing out entire statements (especially when we begin to analyze arguments) can be quite cumbersome.
It is sooooooo convenient to represent statements with symbols.
We often use the letters $p$ and $q$ to stand in for statements.
Using Symbols to Represent Statements
So instead of writing I will paint my bedroom blue, we can simply write $p.$
Moreover, instead of writing I will paint my kitchen green. we can simply write $q.$
Now suppose we wanted to look at the ways we can form compound statements. For example, we can write I will paint my bedroom blue, and I will paint my kitchen green as simply: $p$ and $q.$
Using Symbols to Represent Statements
Connectives also have symbols. For example: the connective and is represented as $\wedge.$
Thus, we can write I will paint my bedroom blue, and I will paint my kitchen green even more succinctly: $$p \wedge q.$$
Symbols For Connectives
$$ \begin{array}{l|l|l} \mbox{Connective} & \mbox{Symbol} & \mbox{Name} \\ \hline p \mbox{ and } q & p \wedge q & \mbox{Conjunction}\\ p \mbox{ or } q & p \vee q & \mbox{Disjunction}\\ p \mbox{ implies } q & p \rightarrow q & \mbox{Conditional}\\ p \mbox{ if and only if } q & p \leftrightarrow q & \mbox{Biconditional}\\ \end{array} $$
Negation
The negation of a statement $p$ is denoted as $\sim p.$
Translating Word Statements to Symbols
Let $p$ be the statement It is raining and $q$ be I will take my umbrella.
Translate the following statements into symbols.
I will not take my umbrella.
$$\sim q$$
It is raining, and I will take my umbrella.
$$p \wedge q$$
If it is raining, I will take my umbrella.
$$p\rightarrow q$$
It is not raining if and only if I don't take my umbrella.
$$\sim p\leftrightarrow \sim q$$
Translating Symbols Back to Word Statements
Let $p$ be the statement It is raining and $q$ be I will take my umbrella. Translate the following symbols back into words:
$\sim p$
It is not raining.
$p \vee q$
It is raining or I will take my umbrella.
$\sim p \rightarrow q$
If it is not raining, then I will take my umbrella.
$q \leftrightarrow p$
I will take my umbrella if and only if it is raining.
Mathematical Aside Stop Taking Notes! The following will NOT be on the exam.
Big Fact: There is a strong connection between logic and sets.
For example, let our universal set $U$ be the collection of all politicians, and $B$ be the collection of all politicians who accept bribes.
The statement Politician $x$ takes bribes can be represented as $x \in B.$
The negation of this statement is $x \notin B.$
Moreover, we could write the above in terms of compliments: $x \in B'$ since $B'$ is the collection of all politicians who don't accept bribes.
Mathematical Aside (Continued) Stop Taking Notes! The following will NOT be on the exam.
Set operations can also be defined in terms of logical connectives: $$A \cup B=\{x| (x \in A) \vee (x \in B) \}$$ $$A \cap B=\{x| (x \in A) \wedge (x \in B) \}$$ Notice the perfect analogy between $\vee$ and $\cup$, as well as $\wedge$ and $\cap.$
Also $(x \in A) \rightarrow (x \in B)$ means precisely that $A \subseteq B.$