Boolean Expressions

A **Boolean expression** is a formula built from **Boolean variables** (letters such as $A$, $B$, $C$), the values $1$ (true) and $0$ (false), and the operators AND ($\land$), OR ($\lor$) and NOT ($\neg$). Just like $3 + 4 \times 2$ in arithmetic, a Boolean expression has exactly one value, but that value is always $1$ or $0$. **The three operators with 1 and 0:** - $\neg A$ flips the value: $\neg 1 = 0$ and $\neg 0 = 1$ - $A \land B$ is $1$ only when both $A$ and $B$ are $1$ - $A \lor B$ is $1$ when at least one of them is $1$ **Order of operations:** first parentheses, then NOT, then AND, then OR. $$(\;) \;\to\; \neg \;\to\; \land \;\to\; \lor$$ So $A \lor B \land C$ means $A \lor (B \land C)$, and $\neg A \land B$ means $(\neg A) \land B$. **Example:** evaluate $A \lor B \land \neg C$ for $A = 0$, $B = 1$, $C = 0$: $$0 \lor 1 \land \neg 0 = 0 \lor 1 \land 1 = 0 \lor 1 = 1$$ An expression with $n$ variables has $2^n$ input combinations. Its **truth table** lists the value of the expression for every one of them, starting with the row where all variables are $0$. Two expressions are **equivalent** when their final columns match in every row. (The Truth Tables lesson started with the all-true row instead; any fixed order works. The interactive truth tables in this lesson also use $1$ and $0$.)

Evaluating with 1 and 0

Evaluate $A \land \neg B$ for $A = 1$ and $B = 0$.

Substitute the values: Replace $A$ with $1$ and $B$ with $0$ = $1 \land \neg 0$

Do NOT first: $\neg 0 = 1$ = $1 \land 1$

Do AND: Both values are $1$, so AND gives $1$ = $1$

Answer: $A \land \neg B = 1$ (true) for $A = 1$, $B = 0$.

AND Before OR

Evaluate $A \lor B \land C$ for $A = 1$, $B = 0$, $C = 0$.

Substitute the values: Replace each letter with its value = $1 \lor 0 \land 0$

Do AND before OR: $0 \land 0 = 0$ = $1 \lor 0$

Do OR: At least one value is $1$ = $1$

Compare with left to right: Working left to right would give $(1 \lor 0) \land 0 = 1 \land 0 = 0$ = A different, wrong result

Answer: $A \lor B \land C = 1$. Working left to right gives $0$, which is wrong because AND comes before OR.

Parentheses Change the Result

Evaluate $\neg(A \land B)$ and $\neg A \land B$ for $A = 0$ and $B = 0$.

First expression: parentheses first: $A \land B = 0 \land 0 = 0$ = $\neg(0)$

Then NOT: $\neg 0 = 1$ = $\neg(A \land B) = 1$

Second expression: NOT applies only to $A$: $\neg A = \neg 0 = 1$ = $1 \land B$

Then AND: $1 \land 0 = 0$ = $\neg A \land B = 0$

Answer: $\neg(A \land B) = 1$ but $\neg A \land B = 0$. The parentheses decide what NOT is applied to.

A Full Truth Table and Equivalence

Build the truth table of $A \lor \neg A \land B$ and compare it with $A \lor B$.

Count the rows: 2 variables, so $2^2 = 4$ rows = $AB$: 00, 01, 10, 11

Helper column $\neg A$: Flip $A$: 0, 0, 1, 1 = $\neg A$: 1, 1, 0, 0

Helper column $\neg A \land B$ (AND before OR): $\neg A$: 1, 1, 0, 0 AND $B$: 0, 1, 0, 1 = 0, 1, 0, 0

Final column $A \lor (\neg A \land B)$: $A$: 0, 0, 1, 1 OR 0, 1, 0, 0 = 0, 1, 1, 1

Column of $A \lor B$ and comparison: $A \lor B$: 0, 1, 1, 1 = Same in all 4 rows

Answer: Both final columns are 0, 1, 1, 1, so $A \lor \neg A \land B$ and $A \lor B$ are equivalent: the shorter expression does the same job.

Three Variables: Where the Parentheses Go

Build the truth tables of $A \land (B \lor C)$ and $A \land B \lor C$. Are they equivalent?

Count the rows: 3 variables, so $2^3 = 8$ rows: 000, 001, 010, 011, 100, 101, 110, 111 = 8 rows

Helper column $B \lor C$: $B \lor C$ is $0$ only when $B = C = 0$ = 0, 1, 1, 1, 0, 1, 1, 1

Final column $A \land (B \lor C)$: $A$ is $1$ only in the last 4 rows = 0, 0, 0, 0, 0, 1, 1, 1

Final column $A \land B \lor C = (A \land B) \lor C$: $A \land B$: 0, 0, 0, 0, 0, 0, 1, 1. OR with $C$: 0, 1, 0, 1, 0, 1, 0, 1 = 0, 1, 0, 1, 0, 1, 1, 1

Compare the columns: They differ in row 2 (001) and row 4 (011) = Not equivalent

Answer: The expressions are not equivalent: for $A = 0$, $B = 0$, $C = 1$ the first gives $0$ and the second gives $1$. One row that differs is enough.

Boolean expressions are the language computers use to make decisions: - **Programming**: every if-condition is a Boolean expression, such as (age at least 12 AND NOT banned) OR has a parent. Forget the parentheses and the program lets in the wrong people - **Search and filters**: online shops and libraries combine filters like in stock AND (red OR blue) - **Spreadsheets**: functions such as AND(), OR() and NOT() decide which rows are highlighted - **Electronics**: every chip turns Boolean expressions into circuits of logic gates, as you will see in the Logic Gates lesson - **Games and smart homes**: a door opens or a light switches on when an expression of several sensors equals 1 If you know the order of operations and can build a truth table, you can predict exactly what a rule does in every possible situation.

Mistake: Working left to right: $1 \lor 0 \land 0 = (1 \lor 0) \land 0 = 0$

Why: AND comes before OR, just as multiplication comes before addition. Without parentheses, the AND part is done first.

Correct: $1 \lor 0 \land 0 = 1 \lor (0 \land 0) = 1 \lor 0 = 1$

Mistake: Applying NOT to everything after it: reading $\neg A \land B$ as $\neg(A \land B)$

Why: Without parentheses, NOT applies only to the variable directly after it.

Correct: $\neg A \land B = (\neg A) \land B$. To negate the whole AND, write $\neg(A \land B)$.

Mistake: Adding like numbers: $1 \lor 1 = 2$

Why: A Boolean expression can only be $1$ or $0$. OR asks whether at least one value is $1$, it does not count them.

Correct: $1 \lor 1 = 1$, $1 \land 1 = 1$ and $\neg 1 = 0$. The answer is always $1$ or $0$.

Mistake: Calling two expressions equivalent after testing only one or two inputs

Why: $A \land (B \lor C)$ and $A \land B \lor C$ give the same value in 6 of the 8 rows, yet they are different.

Correct: Build the full table with $2^n$ rows and compare every row. One row that differs means not equivalent.

Rules in a Program

Programmers write rules as Boolean expressions and test every combination of inputs before the program goes live.

A theme park ride lets you in if $T \land (H \lor P)$: $T$ = you have a ticket, $H$ = you are tall enough, $P$ = a parent rides with you.

Spam Filters

An email program decides with a Boolean expression whether a message goes to the spam folder.

A filter marks an email as spam if $L \lor S \land \neg K$: $L$ = sender is on a block list, $S$ = suspicious words, $K$ = sender is in your contacts.

Smart Home Rules

Two different expressions can control a device in exactly the same way. A truth table shows when a long rule can be replaced by a short one.

A garden light uses the rule $M \lor M \land N$: $M$ = motion detected, $N$ = it is night.

A Boolean expression combines variables, the values 1 (true) and 0 (false) and the operators $\land$, $\lor$, $\neg$

The value of a Boolean expression is always 1 or 0, never 2 or more

Order of operations: parentheses first, then NOT, then AND, then OR

NOT applies only to what comes directly after it: $\neg A \land B$ is not $\neg(A \land B)$

An expression with $n$ variables has a truth table with $2^n$ rows, starting with the row where all variables are 0

Two expressions are equivalent when their final columns are the same in every row

Q: Why does AND come before OR?

A: It is a convention, like multiplication before addition. With 0 and 1, AND behaves like multiplication ($1 \times 0 = 0$), and OR is often compared to addition, so the same order feels natural.

Q: Can I just use parentheses everywhere?

A: Yes. Parentheses that only show the order of operations never change the value, and they make the expression easier to read. $A \lor (B \land C)$ is the same as $A \lor B \land C$, but the first version shows the order at a glance.

Q: Are 1 and 0 the same as true and false?

A: Yes. 1 stands for true and 0 for false. Computers use 1 and 0 because they store everything as bits. The interactive tables in this lesson use 1 and 0 too.

Q: Is there a faster way than a truth table to show that two expressions are equivalent?

A: Yes. Boolean algebra has laws (such as De Morgan's laws) that turn one expression into another step by step. You will learn them in the Boolean Algebra lesson. A truth table is slower but always works.

Exit Presentation

Boolean Expressions

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Logic & Sets / Boolean Logic

Boolean Expressions

Learn to write and evaluate Boolean expressions with 1 and 0, use the order of operations NOT, AND, OR, and build their full truth tables.

Advanced25 min

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