Laws of Logic in Discrete Mathematics
The laws of logic are a set of rules that govern how logical statements can be combined and manipulated. These laws are essential for reasoning and problem-solving in discrete mathematics.
Basic Laws of Logic
The following are some of the most basic laws of logic:
- Commutative laws:
- p ∨ q ≡ q ∨ p
- p ∧ q ≡ q ∧ p
- Associative laws:
- p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r
- p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r
- Identity laws:
- p ∨ 1 ≡ p
- p ∧ 1 ≡ p
- Idempotent laws:
- p ∨ p ≡ p
- p ∧ p ≡ p
- Double negation:
- ¬¬p ≡ p
More Advanced Laws of Logic
The following are some more advanced laws of logic:
- De Morgan's laws:
- ¬(p ∧ q) ≡ ¬p ∨ ¬q
- ¬(p ∨ q) ≡ ¬p ∧ ¬q
- Distributive laws:
- p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
- p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
- Absorption laws:
- p ∨ (p ∧ q) ≡ p
- p ∧ (p ∨ q) ≡ p
- Contradiction law:
- p ∧ ¬p ≡ F
- Tautology law:
- p ∨ ¬p ≡ T
Inverse Laws:
- p ∨ ¬p ≡ T
- p ∧ ¬p ≡ F
Domination Laws:
- p ∨ T ≡ T
- p ∧ F ≡ F
Examples
The following are some examples of how the laws of logic can be used to simplify logical expressions and to prove theorems:
- Simplifying a logical expression:
Sure, here is an all-in-one Markdown file containing the laws of logic in discrete mathematics:
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Laws of Logic in Discrete Mathematics
The laws of logic are a set of rules that govern how logical statements can be combined and manipulated. These laws are essential for reasoning and problem-solving in discrete mathematics.
Basic Laws of Logic
The following are some of the most basic laws of logic:
- Commutative laws:
- p ∨ q ≡ q ∨ p
- p ∧ q ≡ q ∧ p
- Associative laws:
- p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r
- p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r
- Identity laws:
- p ∨ 1 ≡ p
- p ∧ 1 ≡ p
- Idempotent laws:
- p ∨ p ≡ p
- p ∧ p ≡ p
- Double negation:
- ¬¬p ≡ p
More Advanced Laws of Logic
The following are some more advanced laws of logic:
- De Morgan's laws:
- ¬(p ∧ q) ≡ ¬p ∨ ¬q
- ¬(p ∨ q) ≡ ¬p ∧ ¬q
- Distributive laws:
- p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
- p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
- Absorption laws:
- p ∨ (p ∧ q) ≡ p
- p ∧ (p ∨ q) ≡ p
- Contradiction law:
- p ∧ ¬p ≡ F
- Tautology law:
- p ∨ ¬p ≡ T
Inverse Laws:
- p ∨ ¬p ≡ T
- p ∧ ¬p ≡ F
Domination Laws:
- p ∨ T ≡ T
- p ∧ F ≡ F
Examples
The following are some examples of how the laws of logic can be used to simplify logical expressions and to prove theorems:
- Simplifying a logical expression:
Use code with caution. Learn more (p ∨ q) ∧ (p ∨ ¬q) ≡ p ∨ q ∧ ¬q (by the domination laws)
- Proving a theorem:
Theorem: If p → q and q → r, then p → r.
Proof:**
p → q (premise) q → r (premise) ¬p ∨ q (by modus ponens from 1) ¬p ∨ r (by modus ponens from 2 and 3) p → r (by the law of contraposition)