5/9 5. Truth Tables Free Online FlashCards

What does a truth table show?

A truth table lists every possible assignment of truth values to a statement’s basic propositions and gives the resulting value of the entire expression.

What is a truth-functional connective?

A connective is truth-functional when the compound proposition’s truth value is determined entirely by the truth values of its component propositions.

How does negation affect truth value?

Negation reverses a proposition’s truth value: if $p$ is true, $\neg p$ is false; if $p$ is false, $\neg p$ is true.

When is $p \land q$ true?

A conjunction $p \land q$ is true only when both $p$ and $q$ are true; it is false whenever at least one component is false.

How is standard disjunction interpreted?

In standard logic, $p \lor q$ is true when at least one component is true, including the case where both are true. It is false only when both are false.

When is $p \to q$ false?

A conditional $p \to q$ is false only when the antecedent $p$ is true and the consequent $q$ is false. It is true in every other row.

When is $p \leftrightarrow q$ true?

A biconditional $p \leftrightarrow q$ is true when $p$ and $q$ have the same truth value, and false when their values differ.

How many rows does an $n$-proposition table need?

A truth table with $n$ distinct basic propositions requires $2^n$ rows, since each proposition can be true or false.

What is the basic procedure for constructing a truth table?

Construct the table by listing basic propositions, calculating $2^n$ rows, listing every assignment, adding intermediate columns, and evaluating from the inside out.

What is the usual connective precedence?

Using the usual precedence, evaluate $\neg$ before $\land$, then $\lor$, then $\to$, and finally $\leftrightarrow$, unless parentheses change the grouping.

When is $\neg p \lor q$ false?

The final column for $\neg p \lor q$ is false only when $p$ is true and $q$ is false.

What is a tautology?

A tautology is true on every row of its truth table. For example, $p \lor \neg p$ is a tautology.