1 Language, Logic, and Proof

1.1 Introduction to Logic

Objectives:

•

Identify basic components of propositional logic in mathematical phrases
•

Build truth tables to verify truth values of complex propositions

1.1.1 Defining Basic Logic Terms

In our previous section we no doubt ran into issues with proving these number theory statements. How do we begin, how do we end, and what are the important steps to include in between? Let’s begin unpacking these ideas in this section. Mathematicians wish to be as analytic as is reasonable when proving their statements to make their theorems as robust and sturdy as possible. When we are building large structures, we need to make sure that every piece is solid so that the tools we develop are useful. The basics of this analytic language is called propositional logic, which we discuss now.

Definition 1.1.1.

A proposition is a sentence that is either true or false but never both. The truth value of a proposition refers to its attribute of being true or false. (This is also referred to as a statement, like in our readings.)

Typically we refer to propositions by capital letters $A, B$ , etc. In the definitions to come, you can consider $A$ to be a statement like “It is raining outside” and $B$ to be a statement like “The ground is wet”.

Definition 1.1.2.

Let $A$ and $B$ be propositions.

(a)
The proposition “not $A$ ” is true if $A$ is false.
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  This is expressed symbolically as $\neg A$ (LaTeX $\backslash$ neg). The statement $\neg A$ is called the negation of $A$ .
(b)
The proposition “ $A$ and $B$ ” is true if both $A$ and $B$ are true.
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  This is expressed symbolically as $A\wedge B$ (LaTeX $\backslash$ wedge). The statement $A\wedge B$ is called the conjunction of $A$ and $B$ .
(c)
The proposition “ $A$ or $B$ ” if at least one of $A$ or $B$ is true. Even in the case that both $A$ and $B$ is true, the statement “ $A$ or $B$ ” has a truth value of “true”.
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  This is expressed symbolically as $A\vee B$ (LaTeX $\backslash$ vee). The statement $A\vee B$ is called the disjunction of $A$ and $B$ .
(d)
The proposition “If $A$ , then $B$ ” is true if either both $A$ and $B$ are true, or $A$ is false.
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  This is expressed symbolically as $A\Rightarrow B$ (LaTeX $\backslash$ Rightarrow). The statement $A\Rightarrow B$ is called a conditional proposition or an implication (typically the statement reads “ $A$ implies $B$ ”, but there are several more ways to say the same statement).
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  Given the statement above, $A$ is called the hypothesis (or antecedent) and $B$ is called the conclusion (or consequent).
(e)
The proposition “ $A$ if and only if $B$ ” is true if both $A$ and $B$ have the same truth value.
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  This is expressed symbolically as $A\iff B$ (LaTeX $\backslash$ iff or $\backslash$ Leftrightarrow). The statement $A\iff B$ is called a biconditional proposition.
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  If the proposition “ $A\iff B$ ” is true, we say that $A$ and $B$ are logically equivalent. If $A$ is true, $B$ is true, and vice versa. This is one of the strongest statements we can make in logic, and since it comes up so frequently we abbreviate it as “ $A$ iff $B$ ”.

Example 1.1.3.

Let $A$ be the statement “It is raining outside” and $B$ be the statement “The ground outside is wet”. Write each logical statement below in English. Mark which of these statements are true or false at the time you are reading this.

1.

$\neg A$

Blank space for you to write your work.
2.

$A\wedge B$

Blank space for you to write your work.
3.

$\neg A\vee B$

Blank space for you to write your work.
4.

$\neg(\neg A\vee B)$

Blank space for you to write your work.
5.

$A\Rightarrow B$

Blank space for you to write your work.
6.

$A\Leftrightarrow B$

Blank space for you to write your work.

1.1.2 Truth Tables

Consider the following theorem known as the Intermediate Value Theorem from calculus:

If $f$ is a continuous function on a closed interval $[a,b]$ , and $k$ is any number between $f(a)$ and $f(b)$ , then there is at least one number $c$ in the interval $(a,b)$ such that $f(c)=k$ .

One of our goals in having you take this class is enabling you to prove this theorem yourself. We won’t do this this semester, but we will begin working our way up to it. This very first concept already helps: if we make the statement $P$ “ $f$ is a continuous function on a closed interval $[a,b]$ ”, $Q$ is the statement “ $k$ is any number between $f(a)$ and $f(b)$ ,” and $R$ is the statement “there is at least one number $c$ in the interval $(a,b)$ such that $f(c)=k$ ”, then this statement reads

(P\wedge Q)\Rightarrow R.

You may have thought we gave a lot of definitions in the last couple pages. However, notice that we are now able to break down even complicated theorems into simple logical statements. Even complicated propositions can be broken down into simple propositions, each individual statement can have its truth value analyzed, and then we can use these logical operations to rebuild the statement and determine whether the complicated statement is true or false.

But what are the truth values of $P$ , $Q$ , and $R$ that make this statement true? To do this, we form a truth table, which begins by forming all possible combinations of truth values for $P$ , $Q$ , and $R$ ,

Example 1.1.4.

Write out a truth table for the statements $A\wedge B$ and $A\vee B$ .

Example 1.1.5.

A coach promises her players, “If we win tonight, then I will buy you pizza tomorrow.” Determine the cases in which the players can rightly claim to have been lied to.

Using this information, write the truth table for the statement $P\Rightarrow Q$ .

Notice that the only this statement can be proven false is if we find a case where $P$ is true but $Q$ is false. This motivates the following result:

Theorem 1.1.6.

The negation of the statement $P\Rightarrow Q$ is the statement $P\wedge\neg Q$ .

Given this, we can now formulate truth tables for more complicated statements like the one from the previous page:

Example 1.1.7.

Write a truth table for the statement $(P\wedge Q)\Rightarrow R.$ .

Let’s keep investigating the conditional statement $P\Rightarrow Q$ .

Example 1.1.8.

Consider the statement: “If a card has a vowel on one side, it has an even number on the other.”

Given that each card has a letter on one side and a number on the other side, which card(s) do you have to turn over to verify this statement?

T 8 3 A

This seems to motivate the following definition:

Definition 1.1.9.

Given a conditional statement $P\Rightarrow Q$ , the contrapositive of this statement is $\neg Q\Rightarrow\neg P$ .

Example 1.1.10.

Form the contrapositive of the statement “If we win tonight, then I will buy you pizza tomorrow.” If this original statement is true, is the contrapositive of the statement true? How does this connect with our previous example?

Now to prove the following theorem requires a truth table. However, now we are convinced that this theorem might be true:

Theorem 1.1.11.

The statement $P\Rightarrow Q$ and its contrapositive $\neg Q\Rightarrow\neg P$ are logically equivalent. That is, one statement is true if the other is true, and one statement is false if the other is false.

Example 1.1.12.

Prove the theorem above by building a truth table.

There are two other ways to rework the statement “ $P\Rightarrow Q$ ”, but neither of them are logically equivalent to this original statement.

Definition 1.1.13.

Given the statement $P\Rightarrow Q$ , the converse of this statement is $Q\Rightarrow P$ . The inverse of this statement it $\neg P\Rightarrow\neg Q$ .

Example 1.1.14.

Provide an example of a true conditional proposition whose converse is false. Then provide an example of a true conditional proposition whose inverse is false. Could you use the same example for both?