AND, OR: Combining Propositions

Цели урока

• Understand conjunction (AND) and when it is true
• Master disjunction (OR) and its nuances
• Distinguish inclusive from exclusive OR
• Learn to build and read truth tables

"Discount for students OR pensioners" - what about a student who is also a pensioner? Logic and everyday language diverge more often than expected.

• Filters in online stores: "red AND size M"
• Conditions in programming: if (logged_in && has_permission)
• Legal formulations: "a fine or imprisonment"
• Search queries: Google uses AND/OR in different ways

Conjunction: both must be true

**Conjunction** (AND, ∧) is a compound proposition that is true only when **both** simple propositions are true.

**Conjunction notation:** • In logic: P ∧ Q ("P and Q") • In programming: P && Q or P and Q • In mathematics: P · Q • In English: "and", "but" (sometimes), "as well as"

Conjunction is a strict connective. One false component is enough to make the entire proposition false. Like a chain: one weak link - and the whole chain breaks.

Disjunction: at least one is true

**Disjunction** (OR, ∨) is a compound proposition that is true when **at least one** of the simple propositions is true.

**Note:** In logic "OR" is inclusive. "Sea OR mountains" allows both at the same time! This differs from the everyday "either-or" ("tea or coffee?").

Disjunction is a lenient connective. One true component is enough to make the entire proposition true. Like parallel roads: if at least one is open - traffic flows through.

Inclusive OR: logic vs everyday language

In logic **OR is always inclusive**: "A or B" is true even if both are true. This often goes against intuition.

**Why inclusive?** It is simpler for mathematics and programming. Most real-world conditions allow both options: "knowledge of Python OR JavaScript" - knowing both is even better!

When people say "OR" in everyday life, they often mean exclusive: "tea or coffee?" - choose one. But in logic the default is inclusive.

Exclusive OR: one or the other

**Exclusive OR** (XOR, ⊕) is true when **exactly one** of the propositions is true. Both true or both false - the result is false.

**In programming:** • Python: `a ^ b` (for integers) • JavaScript: `a ^ b` • In conditions: `(a || b) && !(a && b)` XOR is used in cryptography, checksums, and toggle switches.

When it matters: "Choose delivery method: courier OR pickup" - this is logically XOR, both cannot be chosen. But the context often makes this implicit.

Truth tables: a cheat sheet for all cases

A **truth table** shows the value of a compound proposition for all possible combinations of values of the simple ones. It is a universal verification tool.

**Why 4 rows?** Two variables × two values = 2² = 4 combinations. For three variables: 2³ = 8 rows. For four: 2⁴ = 16 rows. Tables grow exponentially!

Truth tables are like a calculator for logic. No need to hold everything in mind: just fill in the table and look at the result. In the next lessons we'll build tables for more complex expressions.

Key Takeaways

• Conjunction (AND, ∧): true only when BOTH components are true
• Disjunction (OR, ∨): true when AT LEAST ONE component is true
• In logic OR is inclusive (allows both options)
• XOR (exclusive OR): true when EXACTLY ONE is true

Next

Now we'll study the most important connective - the conditional IF-THEN

• IF-THEN — next lesson
• Validity — previous lesson

Вопросы для размышления

• Find conjunctions and disjunctions in the terms of service of some app. Is OR inclusive or exclusive there?
• How is the difference between AND and OR best explained to a five-year-old?
• In what situations should a programmer use XOR instead of regular OR?

Связанные уроки

• comp-18-type-checking
• ml-01