Do I always need brackets?

With two parts and one operator, no. As soon as AND and OR appear together, or a NOT should apply to more than one statement, brackets make the meaning clear. The rules for reading a statement without brackets (NOT before AND before OR) come in the lesson Boolean Expressions.

Can I use a calculator for truth values?

Partly. With 1 and 0, AND works like multiplication ($1 \times 0 = 0$, $1 \times 1 = 1$) and NOT $P$ equals $1 - P$. The calculator in this lesson does not know the words AND, OR and NOT, so use it for these products and for counting cases like $2^3$.

Why can a question not be part of a compound statement?

A statement must be either true or false. A question like "Is it raining?" or an order like "Close the door!" has no truth value, so it cannot be one of the parts.

Teacher Resources

Compound Boolean Statements

Learn to build compound statements from simple ones with AND, OR and NOT, translate between words and symbols, and find their truth value.

Advanced25 minLesson

Printable worksheet

Definition

A simple statement is one sentence that is either true or false, such as "It is raining." A compound statement joins two or more simple statements with the logical operators AND (

\land

), OR (

\lor

) and NOT (

\neg

). Its truth value depends only on the truth values of its parts.

We write 1 for true and 0 for false. The interactive tables in this lesson use 1 and 0 as well.

From words to symbols. Let

P

: "It is raining" and

Q

: "It is cold".

In words	Symbols	True when
It is raining and it is cold.	$P \land Q$	both parts are true
It is raining or it is cold.	$P \lor Q$	at least one part is true
It is not raining.	$\neg P$	$P$ is false
It is raining but it is not cold.	$P \land \neg Q$	$P$ is true and $Q$ is false
It is neither raining nor cold.	$\neg P \land \neg Q$	both parts are false

Signal words: *and*, *but*, *both ... and* mean AND. *or* means OR, which is also true when both parts are true. *not*, *it is not true that* mean NOT.

Finding the truth value in 3 steps:

1.Give each simple statement its value: 1 or 0.
2.Work out the parts in brackets and every NOT.
3.Combine the results with AND or OR.

The four cases of two statements (starting with 0, like the interactive table; the lesson Truth Tables starts with T, and both orders list the same cases):

$P$	$Q$	$P \land Q$	$P \lor Q$	$\neg P$
0	0	0	0	1
0	1	0	1	1
1	0	0	1	0
1	1	1	1	0

Three parts, eight cases. With a third statement

R

there are

2^3 = 8

cases, and brackets show which parts belong together. For

(P \lor Q) \land \neg R

$P$	$Q$	$R$	$P \lor Q$	$\neg R$	$(P \lor Q) \land \neg R$
0	0	0	0	1	0
0	0	1	0	0	0
0	1	0	1	1	1
0	1	1	1	0	0
1	0	0	1	1	1
1	0	1	1	0	0
1	1	0	1	1	1
1	1	1	1	0	0

This compound statement is true in 3 of the 8 cases: when at least one of

P

and

Q

is true and

R

is false. How to read a statement without brackets (the order of operations) is the topic of the lesson Boolean Expressions.

Try it now

Let

P

: "The dog is hungry" and

Q

: "The dog is tired". Which symbols mean "The dog is hungry and tired"?

Worked Examples

Let $P$ : "The shop is open" and $Q$ : "I have money". Write each sentence in symbols: (a) The shop is open and I have money. (b) The shop is not open or I have money. (c) The shop is open but I have no money. (d) The shop is not open and I have no money.

Sentence (a)

The word *and* joins two parts that are both claimed to be true → $P \land Q$

Sentence (b)

*not* belongs only to the first part, *or* joins the two parts → $\neg P \lor Q$

Sentence (c)

*but* means AND, and *I have no money* is the negation of $Q$ → $P \land \neg Q$

Sentence (d)

Both parts are negated and joined with *and* → $\neg P \land \neg Q$

Answer: (a) $P \land Q$ , (b) $\neg P \lor Q$ , (c) $P \land \neg Q$ , (d) $\neg P \land \neg Q$ . Sentence (d) can also be said with neither ... nor: neither is the shop open nor do I have money.

Common Mistakes

Thinking that " $P$ or $Q$ " is false when both parts are true

Why it's wrong: In everyday speech *or* often means one or the other, but not both. In logic OR is inclusive: $1 \lor 1 = 1$ .

Correct: $P \lor Q$ is false only when both parts are false. For one but not both there is a separate operator, XOR (exclusive OR).

Negating "I like tea and coffee" as "I like neither tea nor coffee"

Why it's wrong: *Neither ... nor* is true only when both parts are false (1 case). The negation of an AND statement must be true whenever at least one part is false (3 cases).

Correct: Say "It is not true that I like both tea and coffee": at least one of them I do not like. The general rule (De Morgan's laws) comes in the lesson Boolean Algebra.

Reading $\neg P \land Q$ as if it were $\neg(P \land Q)$

Why it's wrong: Without brackets NOT applies only to the statement right after it. For $P = 0$ and $Q = 0$ : $\neg P \land Q = 1 \land 0 = 0$ , but $\neg(P \land Q) = \neg 0 = 1$ .

Correct: Check what the NOT touches. Write brackets when NOT should apply to the whole compound statement.

Not recognizing *but* as AND

Why it's wrong: "The test was hard but I passed" sounds like a contrast, yet both parts are claimed to be true.

Correct: Translate *but*, *although* and *both ... and* as AND: $H \land S$ (hard test AND passed).

Interactive Visual

Truth Table

Select Operator

P ∧ Q

P	Q	P ∧ Q
0	0	0
0	1	0
1	0	0
1	1	1

1= True0= False

Truth Table

Expression(P ∨ Q) ∧ ¬R

P	Q	R	P ∨ Q	¬R	(P ∨ Q) ∧ ¬R
0	0	0	0	1	0
0	0	1	0	0	0
0	1	0	1	1	1
0	1	1	1	0	0
1	0	0	1	1	1
1	0	1	1	0	0
1	1	0	1	1	1
1	1	1	1	0	0

1= True0= False

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

18 problems

Problem 1 of 18

Easy

Let $P$ : "The dog is hungry" and $Q$ : "The dog is tired". Which symbols mean "The dog is hungry and tired"?

Why It Matters

Almost every rule you meet is a compound statement:

Online shops and search engines: a filter like "shoes AND size 38 AND NOT red" combines three conditions
Programming: conditions such as (logged in AND NOT blocked) OR admin decide what an app does next
Rules and signs: "Children under 12 enter free if they come with an adult" is met or not depending on two facts
Arguments and debates: knowing exactly when OR and NOT are true helps you spot sentences that sound alike but mean different things
Digital circuits: every chip is built from AND, OR and NOT, the same operators you use here

You already know AND, OR and NOT one at a time. In this lesson you combine them into longer statements and find their truth value with confidence.

Real World Applications

Online Shop Filters

Shops and search engines combine conditions with AND, OR and NOT to decide which items you see.

Example:

The filter (black OR blue) AND NOT used shows only new items, in either of the two colors.

1Try It Yourself

A filter shows a jacket if $(B \lor S) \land C$ , where $B$ : the jacket is blue, $S$ : it is on sale, $C$ : it costs less than 50 dollars. A jacket is blue, is on sale and costs 65 dollars.

Does the filter show this jacket?

Step 1: Write the mathematical expression

With 1 and 0, AND works like multiplication. Write the last step as a product:

Game Rules in Code

Video games and apps check compound conditions many times per second to decide what happens next.

Example:

A door opens if (key AND NOT alarm). With the key and no alarm: $1 \land \neg 0 = 1 \land 1 = 1$ , so the door opens.

2Try It Yourself

A player levels up if $S \land (\neg T \lor K)$ , where $S$ : the score is at least 100, $T$ : the time is over, $K$ : the player has a bonus key.

In how many of the 8 cases does the player level up?

Step 1: Write the mathematical expression

Write the count as the cases with $S$ true minus the cases that fail:

Rules and Signs

Rules on signs, tickets and contracts are compound statements, and it often matters exactly when a rule is NOT met.

Example:

"Free parking on Sundays or after 8 pm" holds on a Sunday at noon, on a Monday at 9 pm, and also on a Sunday at 9 pm.

3Try It Yourself

A museum sign says: children enter free if they are under 12 and come with an adult. Let $U$ : the child is under 12, $A$ : the child comes with an adult.

In how many of the 4 cases is the free entry rule NOT met?

Step 1: Write the mathematical expression

Write the count as all cases minus the cases where the rule is met:

Key Takeaways

1A compound statement joins simple statements with AND ( $\land$ ), OR ( $\lor$ ) and NOT ( $\neg$ )
2*and* and *but* translate to AND, *or* to OR, *not* and *it is not true that* to NOT
3AND is true only when both parts are true; OR is false only when both parts are false
4To find the truth value, give each part 1 or 0, work out the brackets and every NOT, then combine
5Brackets decide what NOT applies to: $\neg(P \land Q)$ is not the same as $\neg P \land Q$
6With 3 simple statements there are $2^3 = 8$ cases, and the negation of an AND statement is true whenever at least one part is false

Frequently Asked Questions

Not always. The operator OR is always inclusive: it is also true when both parts are true. In everyday speech *either ... or* often hints at one but not both. When a sentence clearly means one but not both, write it as the exclusive OR (XOR):

(P \lor Q) \land \neg(P \land Q)

. Otherwise read it as OR.

Glossary

Simple statement: A single sentence that is either true or false, such as " $7$ is prime"
Compound statement: A statement built from simple statements with AND, OR and NOT, such as $P \land \neg Q$
Truth value: Whether a statement is true (1) or false (0)
Conjunction (AND): $P \land Q$ , true only when both parts are true
Disjunction (OR): $P \lor Q$ , true when at least one part is true
Negation (NOT): $\neg P$ , true exactly when $P$ is false

Formula Card

AND (conjunction)

P \land Q

True (1) only when both $P$ and $Q$ are true

OR (disjunction)

P \lor Q

False (0) only when both $P$ and $Q$ are false

NOT (negation)

\neg P

Flips the value: $\neg 1 = 0$ and $\neg 0 = 1$

Number of cases

2^n

$n$ simple statements give $2^n$ cases: 4 for two, 8 for three

Compound Boolean Statements

Definition

Worked Examples

Common Mistakes

Interactive Visual

Truth Table

Truth Table

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Online Shop Filters

Game Rules in Code

Rules and Signs

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

AND (conjunction)

OR (disjunction)

NOT (negation)

Number of cases

More in This Topic

Introduction to Boolean Logic

True and False Statements

The AND Operation

The OR Operation