2.01 Translating logic statements

Logic is the science of reasoning. There are a variety of logical relationships used in math.

Exploration

Use the diagram along with the statements that follow to answer the questions:

• Let p represent the statement "an animal is a corgi."

• Let q represent the statement "an animal is a dog."

• Let r represent the statement "an animal is a cat."

• Let s represent the statement "a corgi is a dog."

1. If an animal is a corgi, must it also be a pet?

2. Determine whether the statement is true or false: \enspace\enspaceIf p, then q.

3. Fill in the blanks with p, q, or r to make a true statement: \enspace \enspaceIf ⬚, then ⬚.

4. Fill in the blanks with p, q, or r to make a true statement: \enspace \enspaceIf it's not true that ⬚, then it's not true that ⬚.

5. What other true statements can you write from the diagram?

In the exploration, p,\,q,\,r, and s are examples of simple logic statements or propositions.

New statements can be formed from these logic statements. Negation is the opposite of a given statement. For the statement r: "an animal is a cat" the negation is "an animal is not a cat."

We can combine simple logic statements to form compound statements. One type of compound statement is a conjuction which connects two statements with the word and. The conjunction of p and q from the exploration is "an animal is a corgi and an animal is a dog."

A disjunction connects two statements with the word or. The disjunction of r and q from the exploration is "an animal is a cat or an animal is a dog." In a disjunction, one or both statements may be true.

A conditional statement, also called an if-then statement, is a type of compound statement that combines two simple statements in the form:

\text{If $p$, then $q$.}

We call p the premise or the hypothesis. The hypothesis is the assertion that begins the statement of argument, and it typically starts with the word "if".

We call q the conclusion. It closes out the argument and is true if the premise is true.

The converse of a conditional is formed by switching the place of the premise and conclusion of the conditional statement:

\text{If $q$, then $p$.}

A true conditional statement can have a false converse, and a false conditional can have a true converse. This pair has a true conditional and a false converse:

This pair has a false conditional and a true converse:

A biconditional statement is the conjunction of a conditional statement and its converse:

\text{If $p$, then $q$; and if $q$, then $p$. }

We may also see this written as:

\text{$p$ if and only if $q$}

Examples

We can use a variety of symbols when working with logical statements. This can make writing logical statements more efficient.

Negation

This is the opposite of a statement. We write the negation of p as \sim p.

Note that the negation of a negation becomes the original proposition. In symbols we could write that \sim \left(\sim p\right) is the same as (or equivalent to) p.

Conjunction

To join two logical statements together with the word and, we use a conjunction which is often symbolized as \land.

Disjunction

To join two logical statements together with the word or, we use a disjunction which is often symbolized as \lor.

Conditionals

The symbol used for the conditional connective is a right arrow \to. Recall a conditional is in the form:

\text{If $p$ then $q$.}

The connective, as a right arrow, tells us that the statement can only be read one way.

Biconditionals

The symbol used for the biconditional connective is a left-right arrow \leftrightarrow. Recall a biconditional is in the form:

p\text{ if and only if } q

This implies both of the following statements are true:

Examples