Logic: Negation, Conjunction, Disjunction & Truth Tables

Geometry runs on statements you can judge. A statement is a sentence that must be either true or false, and that verdict is called its truth value, written $T$ or $F$ . "A pentagon has five sides" is a statement and it is true. "Please hand me the ruler" is not a statement at all, because there is nothing to judge.

Once sentences have truth values, you can combine them. Adding "not" makes a negation, joining two with "and" makes a conjunction, and joining two with "or" makes a disjunction. This page covers what each one means, how to fill in a truth table, and how the same idea looks on a Venn diagram.

Negation: flipping a truth value

The negation of a statement $p$ is written $\sim p$ and read "not $p$ ." It says the opposite of what $p$ says, so it always has the opposite truth value. If $p$ is true, $\sim p$ is false. If $p$ is false, $\sim p$ is true. That is the whole rule, and it never has an exception.

Writing a good negation is where students slip. The negation of "the angle is a right angle" is "the angle is not a right angle," not "the angle is acute." Acute is one specific way to fail at being right, but a negation has to cover every way to fail, obtuse and straight included. When in doubt, insert the word "not" instead of reaching for an opposite word.

Conjunction and disjunction: and is picky, or is generous

A conjunction joins two statements with "and" and is written $p \land q$ . It is true only when both parts are true. One false part sinks the whole thing, which is why "and" is the picky connector.

A disjunction joins two statements with "or" and is written $p \lor q$ . It is true when at least one part is true. Here is the point most students get wrong: in mathematics, "or" always includes the both case. If $p$ is true and $q$ is true, then $p \lor q$ is still true. The only way a disjunction turns out false is when both parts are false.

Truth tables and the Venn picture

A truth table lists every possible combination of truth values for the parts, then works out the truth value of the compound statement in each case. With two statements there are exactly four rows: $TT$ , $TF$ , $FT$ , and $FF$ . Fill the $p$ and $q$ columns in that order every time and you will never miss a case. The $p \land q$ column reads $T$ , $F$ , $F$ , $F$ , and the $p \lor q$ column reads $T$ , $T$ , $T$ , $F$ .

A Venn diagram shows the same information as a picture. Draw one circle for $p$ and an overlapping circle for $q$ . The overlap, where both are true, is the conjunction $p \land q$ . Both circles together, everything inside either one, is the disjunction $p \lor q$ . That picture is also how you count: the number in the overlap belongs to "and," and the whole shaded region belongs to "or."

Worked examples

Example 1: truth values of a compound statement

Let $p$ : A square has four sides. Let $q$ : A square has three sides. Find the truth value of $p \land q$ , $p \lor q$ , and $\sim p \lor q$ .

Judge each part on its own

p = T, \quad q = F

A conjunction needs both parts true, and

q

is false

p \land q = F

A disjunction needs only one true part, and

p

is true

p \lor q = T

Negation flips

p

\sim p = F

Now both parts of the last disjunction are false

\sim p \lor q = F

Answer: $p \land q = F$ , $p \lor q = T$ , and $\sim p \lor q = F$

Example 2: write a negation and give its truth value

Let $r$ : A right angle measures $100^\circ$ . Write $\sim r$ and give the truth value of each.

A right angle measures

90^\circ

, so

r

is false

r = F

Negate by inserting "not," not by naming a different angle

\sim r

: A right angle does not measure

100^\circ

The negation has the opposite truth value

\sim r = T

Answer: $\sim r$ : A right angle does not measure $100^\circ$ ; $r = F$ and $\sim r = T$ .

Example 3: build a truth table for $\sim p \land q$

Construct a truth table for the compound statement $\sim p \land q$ .

List the four combinations of

p

and

q

in order

TT, \; TF, \; FT, \; FF

Fill a

\sim p

column by flipping each

p

value

F, \; F, \; T, \; T

Row 1:

\sim p = F

, so the conjunction fails

F \land T = F

Row 2:

\sim p = F

again

F \land F = F

Row 3: both parts are true

T \land T = T

Row 4:

q

is false

T \land F = F

Answer: The $\sim p \land q$ column reads $F$ , $F$ , $T$ , $F$ — it is true only when $p$ is false and $q$ is true.

Example 4: reading a Venn diagram

In a club, $7$ members play only soccer, $5$ play both soccer and tennis, and $3$ play only tennis. How many play soccer or tennis? How many play soccer and tennis?

"And" is the overlap of the two circles

5

"Or" is everything inside either circle, so add all three regions

7 + 5 + 3

Add

= 15

The

5

in the overlap is counted once, not twice, because "or" includes the both case

Answer: $15$ play soccer or tennis, and $5$ play soccer and tennis.

Try one yourself

Common questions

Does "or" mean one or the other, but not both?

No. In mathematics "or" is inclusive, so $p \lor q$ is true when $p$ is true, when $q$ is true, and when both are true. The only false row of a disjunction is the one where both parts are false. Everyday speech often means the exclusive version, which is why this trips people up.

Is the negation of "the angle is acute" the same as "the angle is obtuse"?

No. The negation is "the angle is not acute," which also covers right angles and straight angles. A negation must be true in every case where the original is false, so name the opposite by adding "not" rather than by picking one other category.

How many rows does a truth table need?

Two statements give $4$ rows, because each of $p$ and $q$ can be $T$ or $F$ . Three statements give $8$ rows. In general $n$ statements give $2^n$ rows. Listing them in a fixed order, all $T$ first down to all $F$ last, keeps you from skipping a case.

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Logic: Negation, Conjunction, Disjunction & Truth Tables

Negation: flipping a truth value

Conjunction and disjunction: and is picky, or is generous

Truth tables and the Venn picture

Worked examples

Example 1: truth values of a compound statement

Example 2: write a negation and give its truth value

Example 3: build a truth table for $\sim p \land q$

Example 4: reading a Venn diagram

Try one yourself

Common questions

Keep learning

Intro to Reasoning: Conditional Statements & Logic

Inductive vs. Deductive Reasoning

Converse, Inverse, Contrapositive & Biconditionals

Logic: Negation, Conjunction, Disjunction & Truth Tables

Negation: flipping a truth value

Conjunction and disjunction: and is picky, or is generous

Truth tables and the Venn picture

Worked examples

Example 1: truth values of a compound statement

Example 2: write a negation and give its truth value

Example 3: build a truth table for ∼p∧q\sim p \land q∼p∧q

Example 4: reading a Venn diagram

Try one yourself

Common questions

Keep learning

Intro to Reasoning: Conditional Statements & Logic

Inductive vs. Deductive Reasoning

Converse, Inverse, Contrapositive & Biconditionals

Example 3: build a truth table for $\sim p \land q$