New 4.4 Logic Basics

Introduction

What you’ll learn to do: Create Venn Diagrams to interpret statements and conditionals

Logic is a systematic way of thinking that allows us to deduce new information from old information and to parse the meanings of sentences. You use logic informally in everyday life and certainly also in doing mathematics. For example, suppose you are working with a certain circle, call it “Circle X,” and you have available the following two pieces of information.

  1. Circle X has radius equal to 3.
  2. If any circle has radius [latex]r[/latex], then its area is [latex]\pi{r}^{2}[/latex] square units.

You have no trouble putting these two facts together to get:

  1. Circle X has area [latex]9\pi[/latex] square units.

You are using logic to combine existing information to produce new information. Since a major objective in mathematics is to deduce new information, logic must play a fundamental role. This chapter is intended to give you a sufficient mastery of logic.

 

Learning Outcomes

  • Determine if a sentence is a statement.
  • Create new statements using connectors and negations.
  • Determine the truth value of a statement using a Venn diagram.
  • Use a Venn diagram to interpret complex statements or conditionals.

 

Logical Statements

Statements

A statement (or proposition) is a sentence that makes a claim that is either true or false.

  • We will often use letters to stand for statements.  For example, the letter P could stand for the statement “An apple is a fruit.”

 

Example

Decide if each of the following sentences are statements.

  1. Whales are mammals.
  2. The Beatles were the best band of the 20th century.
  3. Millie West was the governor of Colorado in 2015.
  4. Why do you like spaghetti?
  5. Please stop doing that!

Try It

 

Negation

The logical opposite of a statement is called its negation.  Note that the negation of a statement will have the opposite truth value of the original statement.

  • If a statement is true, its negation will be false.  If a statement is false, its negation will be true.

Example

Give the negation of the statement: Whales are mammals.

 

Venn Diagrams

We can picture and understand statements by thinking about appropriate sets and Venn diagrams. For instance, if we consider the statement “Maria owns a Honda Accord,” then we can think about the set of all people who own Honda Accords. If Maria happens to be in this set, then the statement is true. A reasonable universal set might be all people. This is depicted in the Venn diagram below.

In this way, we can think about the negation of a statement to be the complement of the relevant set.

 

Compound Statements

Often, we connect two or more statements together to form a new statement. For example, when searching the internet, we use terms like “and” and “or” to help us find specific web pages that fit in the categories we are interested in. When we search for “mystery and comedy” we are looking for a book that satisfies both of these conditions at the same time. If we were to search for “mystery or comedy” we are looking for a book that is a mystery, a comedy, or both.

Logical Connectors

Let P and Q be statements. We can form new statements by connecting them together using the words “and” and “or”.

  • The statement P and Q is called their conjunction. The conjunction is true when both statements P and Q are true. Notice that when thinking about Venn diagrams, the compound statement “P and Q” corresponds to the intersection.
  • The statement P or Q is called their disjunction. The disjunction is true when either statement P is true, OR statement Q is true, OR both statement P and statement Q are true. Notice that when thinking about Venn diagrams, the compound statement “P or Q” corresponds to the union.

Notice here that or is not exclusive. This is a difference between the logic use of the word and common everyday use. When your significant other asks “do you want to go to the park or the movies?” they are usually proposing an exclusive choice – one option or the other, but not both. In mathematical logic, the or is not exclusive – more like being asked at a restaurant “would you like fries or a drink with that?” Answering “both, please” is an acceptable answer.

Example

Consider the statements

  • [latex]P[/latex]: I have been to Denver.
  • [latex]Q[/latex]: I have been to Chicago.

Draw a Venn diagram showing where Nuri would be if [latex]P[/latex] and [latex]Q[/latex] is true for Nuri. Mark any possible regions with an X.

 

Example

Consider the statements

  • [latex]P[/latex]: I have been to Denver.
  • [latex]Q[/latex]: I have been to Chicago.

Draw a Venn diagram showing where June would be if [latex]P[/latex] or [latex]Q[/latex] is true for June. Mark any possible regions with an X.

 

Example

Suppose we are searching a library database for Mexican universities. Express a reasonable search using a logical connector.

 

In most internet search engines, it is not necessary to include the word and; the search engine assumes that if you provide two keywords you are looking for both. In Google’s search, the keyword or has to be capitalized as OR, and a negative sign in front of a word is used to indicate not. Quotes around a phrase indicate that the entire phrase should be looked for. The search from the previous example on Google could be written:

Mexico university -“New Mexico”

 

Example

Describe the numbers that meet the condition:

even and less than 10 and greater than 0

Try It

 

Which Comes First?

Sometimes statements made in English can be ambiguous. For this reason, mathematical logic uses parentheses to show precedent, just like in algebraic order of operations.

The English phrase “Go to the store and buy me eggs and bagels or cereal” is ambiguous; it is not clear whether the requestors is asking for eggs always along with either bagels or cereal, or whether they’re asking for either the combination of eggs and bagels, or just cereal.

For this reason, using parentheses clarifies the intent:

Eggs and (bagels or cereal) means Option 1: Eggs and bagels, Option 2: Eggs and cereal
(Eggs and bagels) or cereal means  Option 1: Eggs and bagels, Option 2: Cereal

 

Example

Describe the numbers that meet the condition:

odd number and less than 20 and greater than 0 and (multiple of 3 or multiple of 5)

Be aware that when a string of conditions is written without grouping symbols, it is often interpreted from the left to right, resulting in the latter interpretation.

Implications

Earlier, we connected two statements using the words “and” and “or” to form new statements. Another way to form new statements is using the structure “if…then…”. In logic, this is called an implication.

An implication “if [latex]P[/latex], then [latex]Q[/latex]” is a logical statement suggesting that a phrase [latex]P[/latex], called the hypothesis or antecedent, implies a consequence [latex]Q[/latex]. Like all logical statements, implications can either be true or false.

Example

Suppose that [latex]P[/latex] is the statement “that shape is a square” and that [latex]Q[/latex] is the statement “that shape has four sides.” We can form two implications using these statements:

  1. One implication that we can form is “If the shape is a square, then the shape has four sides.” In terms of [latex]P[/latex] and [latex]Q[/latex] this can be expressed “If [latex]P[/latex], then [latex]Q[/latex].”
  2. Notice that we could also form the implication “If the shape has four sides, then the shape is a square.” In terms of [latex]P[/latex] and [latex]Q[/latex] this would be expressed “If [latex]Q[/latex], then [latex]P[/latex].”

Only one of the two implications is true. Which is it?

“For all…” and Venn Diagrams

In the previous example, we interpreted the implication “if [latex]P[/latex], then [latex]Q[/latex]” as the statement “all P are also Q.” In general, this can be a helpful way to understand implications and has a nice visual depiction using Venn diagrams.

Example

Consider again the statements [latex]P[/latex]: “that shape is a square,” and [latex]Q[/latex]: “that shape has four sides.” Using a universal set of “All Shapes,” determine appropriate sets which represent the statements and draw a Venn diagram showing their relationship.

Putting together the previous examples, we have three ways to understand implications: as an “if…then…” statement, as an “all…are…” statement, and as subsets.

Key Takeaway

For statements [latex]P[/latex] and [latex]Q[/latex], the following three things are equivalent:

  • If [latex]P[/latex], then [latex]Q[/latex]
  • All [latex]P[/latex] are also [latex]Q[/latex]
  • [latex]P[/latex] is a subset of [latex]Q[/latex]

Example

Consider the implication “If they live in Grand Junction, then they live in Colorado.” Express the implication as:

  1. An “all…are…” statement.
  2. A Venn diagram.

Example

Consider the set of people who have received treatment to recover from alcohol addiction and the set of people who have received treatment to recover from opioids.

  1. Draw a Venn diagram depicting these sets.
  2. Suppose that someone claims “if someone has been treated for opioid addiction, then they have been treated for alcohol addiction.” Does your Venn diagram support this claim?

End of Section 4.4