Truth Tables — AND, OR, NOT, XOR & Implications

A truth table shows every possible combination of truth values for a statement and tells you whether the final result is true or false in each case. If you want to understand AND, OR, NOT, XOR, or implication quickly, a truth table is usually the clearest starting point.

The main operators on this page follow a small set of exact rules:

• $p \land q$ is true only if both are true.
• $p \lor q$ is true if at least one is true.
• $\lnot p$ flips the truth value of $p$.
• $p \oplus q$ is true if exactly one of them is true.
• $p \to q$ is false only when $p$ is true and $q$ is false.

Truth Table for AND, OR, NOT, XOR, and Implication

For two statements $p$ and $q$, there are four possible input rows: $TT$, $TF$, $FT$, and $FF$. A complete truth table has to include all four.

$p$	$q$	$p \land q$	$p \lor q$	$p \oplus q$	$p \to q$	$\lnot p$
T	T	T	T	F	T	F
T	F	F	T	T	F	F
F	T	F	T	T	T	T
F	F	F	F	F	T	T

If you remember only one truth table, this is the one to keep. Most introductory logic questions reduce to reading one of these columns correctly.

What Each Logic Symbol Means

AND Means Both

$p \land q$ is true only when both inputs are true.

That is why the AND column has exactly one true row.

OR Means At Least One

$p \lor q$ is true when one input is true or when both are true.

This is the inclusive meaning of OR. If a problem wants "one or the other, but not both," it should use XOR instead.

NOT Reverses One Statement

$\lnot p$ changes true to false and false to true.

NOT is different from the other operators here because it acts on one statement, not two statements.

XOR Means Exactly One

$p \oplus q$ is true when the inputs differ.

So the middle two rows are true, and the rows where $p$ and $q$ match are false.

Implication Has One False Case

$p \to q$ is false only when $p$ is true and $q$ is false.

That rule can feel strange at first because implication in logic does not mean "causes" in everyday language. It means the claim "if $p$, then $q$" fails only when $p$ happens but $q$ does not.

Worked Example: Why $p \to q$ Is Only False Once

Suppose $p$ means "The number is divisible by 4" and $q$ means "The number is even."

Consider the statement

$$p \to q$$

This means: if a number is divisible by $4$, then it is even.

Now read the four logical cases:

• If $p$ is true and $q$ is true, the statement works.
• If $p$ is true and $q$ is false, the statement fails.
• If $p$ is false, the implication counts as true in propositional logic, because the statement made no promise about cases where the condition did not happen.

That is why $p \to q$ has exactly one false row. In this example, the claim is actually true for every real number because every multiple of $4$ is even.

Common Truth Table Mistakes

• Mixing up OR and XOR. Plain OR includes the case where both inputs are true.
• Reading implication as everyday causation. In a truth table, $p \to q$ is defined by its rows, not by a story about cause and effect.
• Forgetting to list every input combination. With two statements, there must be four rows.
• Treating NOT as a two-input operator. It acts on one statement only.
• Assuming truth tables are only for philosophy or proofs. The same logic appears in Boolean algebra and digital systems.

When Truth Tables Are Used

Truth tables are used to define logical connectives, test whether two statements are equivalent, check whether an argument form is valid, and read Boolean expressions in computing.

They are especially useful when symbolic rules feel abstract. A table forces every case into view, which makes hidden mistakes much easier to catch.

Try a Similar Truth Table

Build the table for

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

Then compare its final column with the column for $p \land \lnot q$. If you want to explore another case after that, try the same process with $p \oplus q$ and see how it differs from plain OR.