**Statements and Quantifiers**Worksheet

**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

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 $A \subseteq B.$