Hi Everyone,
Welcome to My blog,
In this post,Let us understand Propositional Equivalences in a simple way.
Propositional Equivalences:
Propositional Equivalences means they have the exact same truth tables.
Definition:
Two propositions are said to be logically equivalent if they have identical truth tables.
Example:
¬(p ∧ q) ≡ ¬p ∨ ¬q
p. q. p ∧ q. ¬p ∧ q.
T. T. T. F
T. F. F. T
F. T. F. T
F. F. F. T
p. q. ¬p ¬q. ¬p ∨ ¬q
T. T. F. F. F
T F. F. T. T
F. T. T. F. T
F. F. T. T. T
Explanation:
Look at the above example, This proposition¬(p ∧ q) is equivalent to this proposition ¬p ∨ ¬q. Look at the first table of the last column is equivalent to the second table of the last column.So if any of the two propositions of the last column are identical,This is called propositional equivalence.
I hope,you understand the Propositional Equivalences.Now we are going to learn the algebra of propositions.
What is Algebra of Propositions?
Simply say, Algebra of Propositions is method to simplify and manipulate Logical statements using laws.
Now we see the laws:
Name. And form.
Idempotent laws:. p ∧ p ≡ p
Associative laws:. (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Commutative laws: p ∧ q ≡ q ∧ p
Distributive laws:. p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Identity laws:. p ∧ T ≡ p
p ∧ F ≡ F
Complement laws:. p ∧ ¬p ≡ T
¬F ≡ T
DeMorgan’s laws:. ¬(p ∧ q) ≡ ¬p ∨ ¬q
Name. . OR form
Idempotent laws:. p ∨ p ≡ p
Associative laws: (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
Commutative laws: p ∨ q ≡ q ∨ p
Distributive laws: p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
Identity laws: p ∨ F ≡ p
p ∨ T ≡ T
Involution law: ¬¬p ≡ p
Complement laws:. p ∨ ¬p ≡ T
¬T ≡ F
DeMorgan’s laws: ¬(p ∨ q) ≡ ¬p ∧ ¬q
Explanation:
These laws are used to simplify the complex statements and solve logical problems.
Conclusion:
I hope, this post will help you to understand Propositional Equivalences.You should learn the algebra of Propositions laws.Sometimes,In ugc net exam,they will ask these laws in match the following questions.
No comments:
Post a Comment