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VOOZH | about |
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VOOZH | about |
Propositions, Laws, and Algebra is a branch of propositional logic that deals with logical statements, logical operations, and algebraic laws used to simplify and analyze logical expressions. It forms the basis of logical reasoning and is widely used in mathematics and computer science.
Below are the laws of Algebra of Propositions:
- p ∨ p ≅ p
- p ∧ p ≅ p
The truth table of the conjunction and disjunction of a proposition with itself will equal the proposition.
- (p ∨ q) ∨ r ≅ p ∨ (q ∨ r)
- (p ∧ q) ∧ r ≅ p ∧ (q ∧ r)
Associative Law states that propositions also follow associativity and can be written as mentioned above.
- p ∨ (q ∧ r) ≅ (p ∨ q) ∧ (p ∨ r)
- p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r)
Distributive Law states that propositions also follow the distribution and can be written as mentioned above.
- p ∨ q ≅ q ∨ p
- p ∧ q ≅ q ∧ p
It states that propositions follow commutative property i.e if a=b then b=a
- p ∨ T ≅ T
- p ∨ F ≅ p
- p ∧ T ≅ p
- p ∧ F ≅ F
where T is a Tautology, F is a Contradiction and p is a proposition.
In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. In formal language, the rules are written as:
- ~~p ≅ p
- p ∨ ~p ≅ T
- p ∧ ~p ≅ F
- ~T ≅ F
- ~F ≅ T
where T is a Tautology, F is a Contradiction and p is a proposition.
As we know that we can form new propositions using existing propositions and logical connectives. New conditional statements can be formed starting with a conditional statement . In particular, there are three related conditional statements that occur so often that they have special names.
Note : It is interesting to note that the truth value of the conditional statement is the same as it's contrapositive, and the truth value of the Converse of is the same as the truth value of its Inverse. When two compound propositions always have the same truth value, they are said to be equivalent. Therefore,
Example : Let p represent “Today is Friday” and q represent “It is raining today”.
- Implication (p → q): If today is Friday, then it is raining.
- Converse (q → p): If it is raining, then today is Friday.
- Contrapositive (~q → ~p): If it is not raining, then today is not Friday.
- Inverse (~p → ~q): If today is not Friday, then it is not raining.
In natural language, biconditional statements are not always written explicitly using “if and only if”. They are often expressed using “if, then” or “only if” statements where the converse is implied but not directly stated.
For example, the statement “If you complete your homework, then you can go out and play” may also imply “You can go out and play only if you complete your homework.” Thus, such statements can sometimes represent a biconditional relationship.
Logical connectives are used to construct compound propoitions by joining existing propositions. Although parenthesis can be used to specify the order in which the logical operators in the compound proposition need to be applied, there exists a precedence order in Logical Operators. The precedence Order is-
Here, higher the number lower the precedence.
In propositional logic, English sentences can be translated into logical expressions using propositional variables and logical connectives. This helps in analyzing statements and representing them in a precise mathematical form.
Example: “You can access the Internet from campus only if you are a computer science major or you are not a freshman.”
Let:
Using logical connectives, the logical expression for the given statement is:
Some Examples are
If today is Friday, then it is raining.
"You can access the Internet from campus only if you are a computer science major or you are not a freshman."
Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them.
