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Laws of Algebra of Statements
(i) Idempotent Laws
(a) p ∨ p ≡ p
(b) p ∧ p ≡ p
(ii) Associative Laws
(a) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(b) (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
(iii) Commutative Laws
(a) p ∨ q ≡ q ∨ p
(b) p ∧ q ≡ q ∧ P
(iv) Distributive Laws
(a) p ∨ (q A r) ≡ (p ∨ q) ∧ (p ∨ r)
(b) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
(v) De Morgan’s Laws
(a) ~(p ∨ q) ≡ (~ p) ∧ (,_ q)
(b) ~(p ∧ q) ≡ (~ p) ∨ (~ q)
(vi) Identity Laws
(a) p ∧ F ≡ F
(b) p ∧ T ≡ p
(c) p ∨ T ≡ T
(d) p ∨ F ≡ p
(vii) Complement Laws
(a) p ∨ (~ p) ≡ T
(b) p ∧ (~ p) ≡ F
(c) ~ (~p) ≡ p
(d) ~ T ≡ F, ~ F ≡ T
Important Points to be Remembered
(i) The number of rows of table is depend on the number of statements.
(a) If p is false, then ~ p is true.
(b) If P is true, then ~ p is false.
(ii) (a) The converse of p => q is q => p.
(b) The inverse of p => q is ~ p => ~ q.
(iii) The contrapositive of p => q is ~ q => ~ p.
A statement which is neither a tautology nor a contradiction is a contingency.
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FAQs on Laws of Algebra of Statements - Applied Mathematics for Class 11 - Commerce
| 1. What are the laws of algebra of statements? |  |
Ans. The laws of algebra of statements are a set of rules or principles used to manipulate and simplify logical statements. These laws include the commutative, associative, distributive, identity, and negation laws.
| 2. How can the commutative law be applied in algebra of statements? | |
Ans. The commutative law states that the order of the elements in a logical statement does not affect its truth value. In algebra of statements, this means that the order of the logical operators (such as AND or OR) can be changed without changing the overall truth value of the statement.
| 3. Can you explain the distributive law in algebra of statements? | |
Ans. The distributive law in algebra of statements allows us to distribute a logical operator over a set of statements connected by another logical operator. For example, the distributive law allows us to express the statement "A AND (B OR C)" as "(A AND B) OR (A AND C)".
| 4. How does the identity law work in algebra of statements? | |
Ans. The identity law states that a logical statement connected to another statement by the OR operator is true if at least one of the statements is true. Similarly, a logical statement connected to another statement by the AND operator is true only if both statements are true. This law helps simplify and determine the truth value of complex logical statements.
| 5. Can you give an example of how negation is used in algebra of statements? | |
Ans. Negation is a unary operator that reverses the truth value of a statement. In algebra of statements, negation is often used to simplify or negate complex statements. For example, the negation of the statement "A OR B" is "NOT A AND NOT B".
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