Truth Tables & Logical Connectives

• Negation ($\neg$): Represents "not p", $\neg p$ means "not $p$."
  • If $p$ represents "It is raining", then $\neg p$ represents "It is not the case that it is raining".
  • Conventions may vary across different disciplines: symbols like $\sim p$, $\~{p}$ or $!p$ are also used to denote the negation of $p$.
  • If $p$ is true, then $\neg p$ is false; if $p$ is false, then $\neg p$ is true.

$$\begin{array}{c|c} p & \sim p \\ \hline T & F\\ F & T \end{array}$$

• Conjunction ($\land$): Represents "$p$ and $q$". $p \land q$ means "$p$ and $q$."
  • If $p$ represents "It is raining" and $q$ represents "It is soggy", then $p \land q$ represents "It is raining and it is soggy."
  • $p$ but $q$ means the same thing as $p \land q$.
  • $p \land q$ is true only if both $p$ and $q$ are true; otherwise, it is false.

$$\begin{array}{cc|c} p & q & p \land q \\ \hline T & T & T \\ T & F & F \\ F & T & F \\ F & F & F \end{array}$$

• Disjunction ($\lor$): Represents "$p$ or $q$". $p \lor q$ means "$p$ or $q$."
  • If $p$ represents "It is raining" and $q$ represents "It is sunny", then $p \lor q$ represents "It is raining or it is sunny."
  • Neither $p$ nor $q$ means the same thing as $\neg p \land \neg q$.
  • We generally interpret "or" in the inclusive sense, meaning that if at least one of $p$ or $q$ is true, then $p \lor q$ is true. Both can be true as well.

$$\begin{array}{cc|c} p & q & p \lor q \\ \hline T & T & T \\ T & F & T \\ F & T & T \\ F & F & F \end{array}$$

A tautology is a propositional statement that is always true, regardless of the truth values of its components. An example of a tautology is the statement $p \lor \neg p$, which states that either $p$ is true or $p$ is not true. This statement is always true because one of the two possibilities must hold.

A contradiction is a propositional statement that is always false, regardless of the truth values of its components. An example of a contradiction is the statement $p \land \neg p$, which states that $p$ is both true and not true at the same time. This statement is always false because it is impossible for both conditions to hold simultaneously.

$$p \land q \equiv q \land p$$

$$p \lor q \equiv q \lor p$$

$$(p \land q) \land r \equiv p \land (q \land r)$$

$$(p \lor q) \lor r \equiv p \lor (q \lor r)$$

$$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$

$$p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$$

$$p \land t \equiv p$$

$$p \lor c \equiv p$$

$$p \land \neg p \equiv c$$

$$p \lor \neg p \equiv t$$

$$\neg t \equiv c$$

$$\neg c \equiv t$$

$$p \land p \equiv p$$

$$p \lor p \equiv p$$

$$p \land (p \lor q) \equiv p$$

$$p \lor (p \land q) \equiv p$$