Proposition

• a sentence that declares a fact
• either true or false but not both

Negation

Definition 1. Let \(p\) be a proposition. The negation of \(\mathbf{p}\), denoted by \(\mathbf{\neg{p}}\) (also denoted by \(\mathbf{\overline{p}}\)), is the statement

“It is not the case that \(\mathbf{p}\).”

• The proposition \(\mathbf{\neg{p}}\) is read “not \(\mathbf{p}\)”
• The truth value of the negation of \(\mathbf{p}\), \(\mathbf{\neg{p}}\).

Truth Table : Negation

Exercises, page 16

3. What is the negation of each of these propositions?

   1. Today is Thursday.
   2. There is no pollution in New Jersey.
   3. 2 + 1 = 3.
   4. The summer in Maine is hot and sunny.

Conjunction

Definition 2. Let \(p\) and \(q\) be propositions. The conjunction of \(p\) and \(q\), denoted by \(p \wedge q\), is the proposition

“\(p\) and \(q\)”.

• The conjunction of \(p \wedge q\) is true when both \(p\) and \(q\) are true and is false otherwise.

Truth Table : Conjunction

Disjunction

Definition 3. Let \(p\) and \(q\) be propositions. The disjunction of \(p\) and \(q\), denoted by \(p \vee q\), is the proposition

“\(p\) or \(q\)”.

• The disjunction \(p \vee q\) is false when both \(p\) and \(q\) are false and is true otherwise.

Truth Table : Disjunction

Exercise 1, page 16.

4. Let \(p\) and \(q\) be the propositions
   • \(p\) : I bought a lottery ticket this week.
   • \(q\) : I won the million dollar jackpot on Friday.

Express each of these propositions as an English sentence.

1. \(\neg{p}\)

2. \(p \vee q\)

3. \(p \wedge q\)

4. \(\neg{p} \wedge \neg{q}\)

5. \(\neg{p} \vee (p \wedge q)\)