Connectives - words such as and, or, if, then

Exclusive or - one or the other of the given events can happen, but not both

Inclusive or - one or the other or both of the given events can happen

*In this class, we will use the inclusive or

Statement - A sentence that can be judged either true or false. Labeling a statement true or false is called assigning a truth value to the statement.

Simple Statements - Sentences that convey only one idea and can be assigned a truth value.

Compound Statements - Sentences that combine two or more simple statements and can be assigned a truth value.

Negation of a statement – change a statement to its opposite meaning.

• The negation of a false statement is always a true statement.

• The negation of a true statement is always a false statement.

Examples:

The moon is made of green cheese. It is not true that the moon is made of green cheese.

All chickens are roosters. It is not true that all chickens are roosters.

Quantifiers - words such as all, none, no, some, etc… Be careful when negating statements that contain quantifiers.

Negation of Quantified Statements

Form of statement Form of Negation

All are. Some are not.

None are. Some are.

Some are. None are.

Some are not. All are.

Example 1: Write Negations

Write the negation of the statement: All houses have two stories.

Solution: This is a false statement, since some houses have one story, some three or more. The negation “Some houses do not have two stories” or “Not all houses have two stories” or “At least one house does not have two stories” are all true statements.

Compound Statements: Statements consisting of two or more simple statements are called compound statements. The connectives often used to join two simple statements are and, or, if…then…, and if and only if.

Not Statements (Negation): The symbol used in logic to show the negation of a statement is ~. It is read “not”. The negation of p is: ~ p.

And Statements (Conjunction): ⋀ is the symbol for a conjunction and is read “and.” The conjunction of p and q is: p ⋀ q. The other words that may be used to express a conjunction are: but, however, and nevertheless.

Example 2: Write a Conjunction: Write the following conjunction in symbolic form:

Green Day is not on tour, but Green Day is recording a new CD.

Solution: Let t and r represent the simple statements.

t: Green Day is on tour.

r: Green Day is recording a new CD.

In symbolic form, the compound statement is ~t ⋀ r.

Or Statements (Disjunction): The disjunction is symbolized by ⋁ and read “or.”

In this book the “or” will be the inclusive or (except where indicated in the exercise set).

The disjunction of p and q is: p ⋁ q.

Example 3: Write a Disjunction

Let

p: Maria will go to the circus.

q: Maria will go to the zoo.

Write the statement in symbolic form.

Maria will go to the circus or Maria will go the zoo.

Solution: p ⋁ q

Let

p: Maria will go to the circus.

q: Maria will go to the zoo.

Write the statement in symbolic form.

Maria will go to the zoo or Maria will not go the circus.

Solution: q ⋁ ~p

Compound Statements: When a compound statement contains more than one connective, a comma can be used to indicate which simple statements are to be grouped together.

When we write the compound statement symbolically, the simple statements on the same side of the comma are to be grouped together within parentheses.

Example 4: Understand How Commas Are Used to Group Statements

Let p: Dinner includes soup.

q: Dinner includes salad.

r: Dinner includes the vegetable of the day

Write the statement in symbolic form.

Dinner includes soup, and salad or vegetable of the day.

Solution: p ⋀ (q ⋁ r)

Example 5: Change Symbolic Statements into Words

Let p: The house is for sale.

q: We can afford to buy the house.

Write the symbolic statement in words.

p ⋀ ~q

Solution: The house is for sale and we cannot afford to buy the house.

Write the symbolic statement in words.

~p ⋁ ~q

Solution: The house is not for sale or we cannot afford to buy the house.

Write the symbolic statement in words.

~(p ⋀ q)

Solution: It is false that the house is for sale and we can afford to buy the house.

If-Then Statements: The conditional is symbolized by → and is read “if-then.” The antecedent is the part of the statement that comes before the arrow. The consequent is the part that follows the arrow. If p, then q is symbolized as: p → q.

Example 6: Write Conditional Statements:

Let p: The portrait is a pastel.

q: The portrait is by Beth Anderson.

Write the statement symbolically.

If the portrait is a pastel, then the portrait is by Beth Anderson.

Solution: p → q

If the portrait is by Beth Anderson, then the portrait is not a pastel.

Solution: q → ~p

It is false that if the portrait is by Beth Anderson, then the portrait is a pastel.

Solution: ~(q → p)

If and Only If Statements: The biconditional is symbolized by ↔ and is read “if and only if.”

If and only if is sometimes abbreviated as “iff.”

The statement p ↔ q is read “p if and only if q.”

Example 7: Write Statements Using the Biconditional

Let p: Alex plays goalie on the lacrosse team.

q: The Titans win the Champion’s Cup.

Write the symbolical statement in words.

p ↔ q

Solution: Alex plays goalie on the lacrosse team if and only if the Titans win the Champion’s Cup.

Write the symbolical statement in words.

q ↔ ~p

Solution: The Titans win the Champion’s cup if and only if Alex does not play goalie on the lacrosse team.

Write the symbolical statement in words.

~(p ↔ ~q)

Solution: It is false that Alex plays goalie on the lacrosse team if and only if the Titans do not win the Champion’s Cup.

Logical Connectives (Summary):

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