// reference · AND
∧ Conjunction_
Conjunction joins two propositions with the word "and". The expression p ∧ q is true only when both p and q are true at the same time; in every other case it is false. It is the strictest connective in propositional logic.
Truth table: p ∧ q
| p | q | p ∧ q★ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Classification: Contingency · 4 rows
Definition
The conjunction of p and q, written p ∧ q, is true only when both propositions are true. If either one is false, the whole conjunction is false.
You will meet several notations: p ∧ q (logic), p & q (computer science), p · q or simply pq (Boolean algebra), and p AND q (electronics). The calculator accepts ∧, &, · and *.
How to read it
p ∧ q is read "p and q". In natural language it also appears as "p but q", "p although q", "both p and q", or "p, moreover q" — all of them assert both parts at once, so they translate to a conjunction.
For example, "Anna studies and works" becomes p ∧ q with p = "Anna studies" and q = "Anna works".
When it is true
The table has four rows. With p = T and q = T, p ∧ q = T. With p = T and q = F, p ∧ q = F. With p = F and q = T, p ∧ q = F. With p = F and q = F, p ∧ q = F.
Only one of the four combinations is true, so p ∧ q is a contingency. A handy memory aid: conjunction behaves like multiplying zeros and ones (1·1 = 1, everything else gives 0).
Everyday example
To pass a course you must "submit the project and pass the exam". If you submit the project but fail the exam, the full requirement is not met. Only doing both satisfies the conjunction.
Programming works the same way: the condition if (age >= 18 && hasLicense) runs only when both comparisons are true.
Properties and equivalences
Commutative: p ∧ q ≡ q ∧ p. Associative: (p ∧ q) ∧ r ≡ p ∧ (q ∧ r). Idempotent: p ∧ p ≡ p. Identity element: p ∧ 1 ≡ p. Annihilator: p ∧ 0 ≡ 0.
Distributive over disjunction: p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r). Absorption: p ∧ (p ∨ q) ≡ p.
Equivalences with other operators: p ∧ q ≡ ¬(¬p ∨ ¬q) (De Morgan), p ∧ q ≡ ¬(p ⇒ ¬q), and p ∧ q ≡ ¬(p ⊼ q) — conjunction is the negation of NAND.
Common mistakes
Assuming "p but q" or "p although q" are not conjunctions. Logically they are: the contrastive flavour does not change the truth value.
Misapplying De Morgan: ¬(p ∧ q) is equivalent to ¬p ∨ ¬q, not ¬p ∧ ¬q. And mixing up precedence: in p ∧ q ∨ r the conjunction is evaluated first, so it means (p ∧ q) ∨ r.
Equivalent expressions
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related laws and rules
Logical equivalence
De Morgan's law (conjunction)
¬(p ∧ q) ⇔ (¬p ∨ ¬q)
Logical equivalence
Distributive law of ∧ over ∨
(p ∧ (q ∨ r)) ⇔ ((p ∧ q) ∨ (p ∧ r))
Logical equivalence
Absorption law
(p ∨ (p ∧ q)) ⇔ p
Logical equivalence
Idempotent law
(p ∧ p) ⇔ p
Logical equivalence
Commutative law
(p ∧ q) ⇔ (q ∧ p)
Fundamental law
Law of non-contradiction
¬(p ∧ ¬p)
Frequently asked questions
How many true rows does p ∧ q have? ▼
Just one: when p and q are both true. The other three combinations are false.
Are "but" and "although" also conjunctions? ▼
Yes. Sentences like "it rains but it is warm" assert both parts, so they formalize as p ∧ q. The contrast is a nuance of language, not of logic.
How is conjunction related to NAND? ▼
They are opposites: p ⊼ q ≡ ¬(p ∧ q). That is why conjunction can be written using NAND alone: p ∧ q ≡ (p ⊼ q) ⊼ (p ⊼ q).