Boolean logic: AND, OR and NOT

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レッスンノート

What is Boolean Logic?

Boolean logic is a branch of algebra that deals with truth values: true and false, often represented as 1 and 0.
Unlike normal algebra, Boolean variables are not numbers but truth values.
Boolean logic uses logical operators (AND, OR, NOT) instead of arithmetic operators like + and −.
It is named after George Boole, who introduced it in the 19th century.
Boolean logic is fundamental to digital electronics and programming.

The Three Basic Operators

AND (∧): The result is true only if both inputs are true.
OR (∨): The result is true if at least one input is true.
NOT (¬): The result is the opposite of the input (true becomes false, false becomes true).
These operators can be combined to form Boolean expressions.

Truth Tables

A truth table shows the output for every possible combination of inputs.
For AND: true AND true = true; all other combinations = false.
For OR: false OR false = false; all other combinations = true.
For NOT: true becomes false, false becomes true.
Truth tables are used to test and verify Boolean expressions.

Evaluating Boolean Expressions

Expressions are evaluated by applying the operators to the input values.
Order of operations: NOT is applied first, then AND, then OR (unless brackets are used).
Example: NOT true AND false → NOT true = false, then false AND false = false.
Brackets can change the order: NOT (true AND false) → true AND false = false, then NOT false = true.
Practice with simple examples to become confident.

Uses in Programming

Boolean logic is used in conditions (if statements) to control program flow.
Example: `if (age ≥ 18 AND hasID) then allow entry`.
Loops use Boolean conditions to decide when to repeat.
Boolean variables can store true/false values (e.g., `isRaining = true`).
Programming languages provide logical operators like `&&` (AND), `||` (OR), and `!` (NOT).

Uses in Search Queries

Search engines use Boolean logic to refine searches.
AND narrows results: `cats AND dogs` finds pages with both words.
OR broadens results: `cats OR dogs` finds pages with either word.
NOT excludes terms: `cats NOT dogs` finds pages about cats but not dogs.
Many search engines support these operators in advanced search.

History and Applications

Boolean algebra was introduced by George Boole in 1847.
Claude Shannon applied it to switching circuits in the 1930s, leading to modern digital logic.
Boolean logic is used in circuit design, set theory, and statistics.
It underpins all digital computers and logic gates.
The Boolean satisfiability problem (SAT) is a key problem in computer science.

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練習問題

無料プレビュー — 56問中8問。すべて見るには登録を。

1.Which of the following is a Boolean value?
Easy
- A1
- B0
- Ctrue
- Dall of the above
2.Which of the following is the Boolean operator for AND?
Easy
- AOR
- BNOT
- CAND
- DXOR
3.In Boolean algebra, the value 1 represents true and 0 represents false.
Easy
True or false?
4.What is the result of the Boolean expression: TRUE AND FALSE?
Easy
- ATRUE
- BFALSE
- C1
- D0
5.What is the result of the Boolean expression: TRUE OR FALSE?
Easy
- ATRUE
- BFALSE
- C1
- D0
6.What is the result of the Boolean expression: NOT TRUE?
Easy
- ATRUE
- BFALSE
- C1
- D0
7.The Boolean operator NOT is a unary operator, meaning it takes only one operand.
Easy
True or false?
8.Which of the following Boolean expressions evaluates to TRUE?
Medium
- ATRUE AND FALSE
- BFALSE OR FALSE
- CNOT FALSE
- DTRUE AND TRUE AND FALSE

もっと問題

9.The expression (TRUE AND TRUE) OR FALSE evaluates to TRUE.
Easy
True or false?
10.In a computer program, which Boolean operator would you use to check that a number is greater than 5 AND less than 10?
Medium
- AOR
- BAND
- CNOT
- DXOR
11.Which Boolean operator would you use in a search engine to find pages that contain both 'cats' and 'dogs'?
Medium
- AOR
- BAND
- CNOT
- DNEAR
12.Which of the following are Boolean operators? (select all that apply)
Medium
- AAND
- BOR
- CNOT
- DPLUS
- EMINUS
13.In Boolean algebra, the expression (A AND B) OR (NOT A) is always true regardless of the values of A and B.
Medium
True or false?
14.Match each Boolean operator to its symbol.
Medium
- AND
- OR
- NOT
- ∧
- ∨
- ¬
15.Put the following steps in the correct order to evaluate the expression: (TRUE OR FALSE) AND NOT FALSE
Medium
- Evaluate NOT FALSE
- Evaluate TRUE OR FALSE
- Evaluate the AND
16.Boolean algebra is used in the design of digital circuits.
Easy
True or false?
17.Who is credited with introducing Boolean algebra?
Easy
- AClaude Shannon
- BGeorge Boole
- CGottfried Leibniz
- DCharles Babbage
18.In Boolean algebra, the operation XOR is the same as OR.
Easy
True or false?
19.Which of the following expressions is equivalent to NOT (A AND B)?
Medium
- ANOT A AND NOT B
- BNOT A OR NOT B
- CNOT A OR B
- DA AND NOT B
20.The Boolean expression (A OR B) AND (NOT A) is equivalent to B.
Medium
True or false?
21.In programming, which of the following is a common symbol for the AND operator?
Medium
- A&&
- B||
- C!
- D==
22.Boolean algebra is used in set theory and statistics.
Easy
True or false?
23.Which of the following is NOT a Boolean value?
Easy
- Atrue
- Bfalse
- C1
- D2
24.The expression (TRUE AND TRUE) OR (FALSE AND FALSE) evaluates to TRUE.
Easy
True or false?
25.Which Boolean expression correctly represents the condition 'x is not greater than 10'?
Medium
- Ax > 10
- BNOT (x > 10)
- Cx < 10
- Dx ≥ 10
26.In Boolean algebra, the expression (A AND B) OR (A AND NOT B) is equivalent to A.
Medium
True or false?
27.Which of the following is a Boolean operator?
Easy
- AAND
- BPLUS
- CMINUS
- DTIMES
28.The Boolean expression (FALSE OR TRUE) AND (TRUE OR FALSE) evaluates to TRUE.
Easy
True or false?
29.Which of the following expressions is equivalent to NOT (A OR B)?
Medium
- ANOT A AND NOT B
- BNOT A OR NOT B
- CNOT A OR B
- DA AND NOT B
30.In a search engine, which Boolean operator would you use to exclude a term from your search?
Medium
- AAND
- BOR
- CNOT
- DNEAR
31.The expression (A OR B) AND (NOT A) is equivalent to B AND NOT A.
Medium
True or false?
32.Which of the following is the correct order of precedence for Boolean operators (from highest to lowest)?
Easy
- ANOT, AND, OR
- BAND, OR, NOT
- COR, AND, NOT
- DNOT, OR, AND
33.Match each Boolean operator to its common symbol.
Easy
- AND
- OR
- NOT
- ∧
- ∨
- ¬
34.Put the following steps in the correct order to evaluate the expression: (TRUE OR FALSE) AND NOT FALSE.
Medium
- Evaluate NOT FALSE
- Evaluate TRUE OR FALSE
- Evaluate AND
35.A school's online safety rule states: 'You may only use the internet if you have permission AND you are supervised.' If a student has permission but is not supervised, what is the outcome?
Easy
- AAllowed to use the internet
- BNot allowed to use the internet
- CAllowed only for educational sites
- DAllowed if they ask again
36.A game character can jump if it is 'on the ground' OR 'has a double-jump power-up'. If the character is in the air and does NOT have the power-up, can it jump?
Easy
- AYes, because OR only needs one condition true
- BNo, because both conditions are false
- CYes, because it is in the air
- DNo, because it is on the ground
37.A smart home system turns on the lights if it is dark AND motion is detected. One evening it is dark but no motion is detected. Which of the following correctly represents the situation?
Medium
- Adark = TRUE, motion = FALSE, lights = TRUE
- Bdark = TRUE, motion = FALSE, lights = FALSE
- Cdark = FALSE, motion = TRUE, lights = TRUE
- Ddark = TRUE, motion = TRUE, lights = FALSE
38.A website allows access if the user is logged in AND (has a premium account OR is an admin). Which of the following combinations of conditions will allow access? (Select all that apply)
Medium
- Alogged in = TRUE, premium = FALSE, admin = FALSE
- Blogged in = TRUE, premium = TRUE, admin = FALSE
- Clogged in = TRUE, premium = FALSE, admin = TRUE
- Dlogged in = FALSE, premium = TRUE, admin = TRUE
- Elogged in = TRUE, premium = TRUE, admin = TRUE
39.In Boolean logic, the expression (TRUE AND FALSE) OR NOT FALSE evaluates to TRUE.
Easy
True or false?
40.Match each Boolean expression to its equivalent simplified form.
Medium
- A AND (B OR NOT B)
- A OR (A AND B)
- NOT (A AND NOT A)
- A
- TRUE
41.Complete the sentence.
Easy
In Boolean logic, the two possible values are ____ and ____.
42.How many possible combinations of inputs are there for a Boolean expression with 3 variables?
Medium
43.What is the result of the expression: (TRUE AND FALSE) OR (NOT FALSE)?
Medium
44.What is the output of the Boolean expression: (TRUE OR FALSE) AND (NOT FALSE)?
Easy
45.What is the result of the expression: NOT (FALSE OR TRUE)?
Medium
46.What is the output of the Boolean expression: (NOT TRUE) OR (FALSE AND TRUE)?
Easy
47.What is the result of the expression: (TRUE AND FALSE) OR (TRUE OR FALSE)?
Medium
48.What is the output of the Boolean expression: (FALSE OR FALSE) AND (TRUE OR TRUE)?
Easy
49.What is the result of the Boolean expression: (TRUE AND FALSE) OR (NOT FALSE)?
Medium
50.A vending machine gives a free drink if the customer inserts a valid coupon OR pays the full price. Write a Boolean expression for this condition using variables C (coupon valid) and P (paid full price).
Medium
51.A Boolean expression has three input variables, A, B, and C. How many rows are there in its truth table?
Medium
52.Complete the sentence about the NOT operator.
Medium
The NOT operator takes a single input and gives the ____ value.
53.Explain the difference between the Boolean operators AND and OR.
Medium
54.Explain how the Boolean operator OR differs from AND, using an example from everyday life.
Hard
55.How many possible combinations of inputs are there for a Boolean expression with 3 variables?
Medium
- A8
- B10
- C7
- D9
56.A Boolean expression has three input variables, A, B, and C. How many rows are there in its truth table?
Medium
- A8 rows
- B7 rows
- C10 rows
- D9 rows

復習ノート

## Boolean Logic: AND, OR, NOT Boolean logic is a branch of algebra that deals with **truth values**: `true` (1) and `false` (0). Unlike normal algebra, it uses logical operators instead of arithmetic ones. It was introduced by **George Boole** in 1847 and is fundamental to digital electronics and programming. ### Basic Operations - **AND (∧)**: Returns `true` only if **both** inputs are `true`. - `true ∧ true = true` - `true ∧ false = false` - `false ∧ true = false` - `false ∧ false = false` - **OR (∨)**: Returns `true` if **at least one** input is `true`. - `true ∨ true = true` - `true ∨ false = true` - `false ∨ true = true` - `false ∨ false = false` - **NOT (¬)**: Flips the value: `true` becomes `false`, and `false` becomes `true`. - `¬true = false` - `¬false = true` ### Truth Tables A truth table shows all possible outputs for a logical expression. For example, for inputs A and B: | A | B | A AND B | A OR B | NOT A | |---|---|---------|--------|-------| | 0 | 0 | 0 | 0 | 1 | | 0 | 1 | 0 | 1 | 1 | | 1 | 0 | 0 | 1 | 0 | | 1 | 1 | 1 | 1 | 0 | ### Evaluating Expressions Use the order: **NOT** first, then **AND**, then **OR** (unless brackets say otherwise). Example: `true OR (false AND NOT true)` 1. `NOT true = false` 2. `false AND false = false` 3. `true OR false = true` So the result is `true`. ### Uses in Programming and Search - **Programming**: Conditions like `if (age >= 18 AND hasID)` control what code runs. - **Search queries**: Use AND, OR, NOT to filter results (e.g., `cats AND dogs` finds pages with both). ### Key Points - Boolean values are `0` and `1`, not numbers. - Boolean algebra is the basis of logic gates in circuits. - It helps in designing digital electronics and in programming logic.

Boolean algebra: A branch of algebra dealing with truth values (true/false, 1/0) and logical operators (AND, OR, NOT), introduced by George Boole.
Truth value: Either true (1) or false (0).
AND (∧): Logical operator that returns true only if both operands are true.
OR (∨): Logical operator that returns true if at least one operand is true.
NOT (¬): Logical operator that negates a value: true becomes false, and false becomes true.
Truth table: A table showing all possible input combinations and the corresponding output for a logical expression.
Logic gate: An electronic circuit that implements a Boolean operation (e.g., AND, OR, NOT).

フラッシュカード

What are the two truth values in Boolean logic?

True and false, usually denoted by 1 and 0.

What is the Boolean operator AND also called?

Conjunction, denoted as ∧.

What is the Boolean operator OR also called?

Disjunction, denoted as ∨.

What is the Boolean operator NOT also called?

Negation, denoted as ¬.

What is the result of TRUE AND FALSE?

FALSE.

What is the result of TRUE OR FALSE?

TRUE.

What is the result of NOT TRUE?

FALSE.

What is the result of NOT FALSE?

TRUE.

What is the result of TRUE AND TRUE?

TRUE.

What is the result of FALSE OR FALSE?

FALSE.

In Boolean algebra, what do 1 and 0 represent?

They represent the truth values true and false, respectively.

Who introduced Boolean algebra and in which book?

George Boole introduced it in his book 'The Mathematical Analysis of Logic' (1847).

How does Boolean algebra differ from elementary algebra?

Boolean algebra uses truth values (true/false) and logical operators (AND, OR, NOT), while elementary algebra uses numbers and arithmetic operators.

What is the Boolean satisfiability problem (SAT)?

It is the problem of determining whether the variables of a given Boolean formula can be assigned to make the formula evaluate to true.

What is a truth table?

A table that shows all possible input combinations and the corresponding output for a Boolean expression.

What is the output of an AND gate when both inputs are 1?

1 (true).

What is the output of an OR gate when at least one input is 1?

1 (true).

What is the output of a NOT gate when the input is 1?

0 (false).

How is Boolean logic used in programming conditions?

It is used to combine conditions with AND, OR, and NOT to control the flow of a program.

How is Boolean logic used in search queries?

It allows combining keywords with AND, OR, and NOT to refine search results.

What is the result of (TRUE AND FALSE) OR TRUE?

TRUE, because TRUE AND FALSE is FALSE, and FALSE OR TRUE is TRUE.

What is the result of NOT (TRUE OR FALSE)?

FALSE, because TRUE OR FALSE is TRUE, and NOT TRUE is FALSE.

What is the result of (NOT FALSE) AND TRUE?

TRUE, because NOT FALSE is TRUE, and TRUE AND TRUE is TRUE.

What is the result of (TRUE OR TRUE) AND (FALSE OR FALSE)?

FALSE, because TRUE OR TRUE is TRUE, FALSE OR FALSE is FALSE, and TRUE AND FALSE is FALSE.

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