This article explores fundamental laws and concepts in the algebra of propositions like Idempotent, Associative, Distributive, and Commutative Laws, as well as special conditional statements.
Laws of Algebra of Propositions
Below mentioned are the laws of Algebra of Propositions
Idempotent Law
- p ∨ p ≅ p
- p ∧ p ≅ p
The truth table of conjunction and disjunction of a proposition with itself will equal the proposition.
Associative Law
- (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.
Distributive Law
- 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.
Commutative Law
- p ∨ q ≅ q ∨ p
- p ∧ q ≅ q ∧ p
It states that propositions follow commutative property i.e if a=b then b=a
Identity Law
- 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.
De Morgan’s Law
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 –
- [Tex]\neg (p\wedge q) \equiv \neg p \vee \neg q
[/Tex]
- [Tex]\neg (p\vee q) \equiv \neg p \wedge \neg q
[/Tex]
Proof by Truth Table
[Tex]\begin{tabular}{ ||c||c||c||c||c||c||c||c||c||c|| } \hline p & q & \neg p & \neg q & p\wedge q & \neg p\vee \neg q & p\vee q & \neg p\wedge \neg q \\ \hline T & T & F & F & T & F & T & F \\ \hline T & F & F & T & F & T & T & F \\ \hline F & T & T & F & F & T & T & F \\ \hline F & F & T & T & F & T & F & T \\ \hline\end{tabular}[/Tex]
Involution Law
Complement Law
- p ∨ ~p ≅ T
- p ∧ ~p ≅ F
- ~T ≅ F
- ~F ≅ T
where T is a Tautology, F is a Contradiction and p is a proposition.
Special Conditional Statements
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 [Tex]p\rightarrow q
[/Tex]. In particular, there are three related conditional statements that occur so often that they have special names.
- Implication : [Tex]p\rightarrow q
[/Tex]
- Converse : The converse of the proposition [Tex]p\rightarrow q
[/Tex]is [Tex]q\rightarrow p
[/Tex]
- Contrapositive : The contrapositive of the proposition [Tex]p\rightarrow q
[/Tex]is [Tex]\neg q\rightarrow \neg p
[/Tex]
- Inverse : The inverse of the proposition [Tex]p\rightarrow q
[/Tex]is [Tex]\neg p\rightarrow \neg q
[/Tex]
To summarise,
[Tex]\begin{tabular}{ ||c||c|| } \hline Statement & If p, then q\\ \hline \hline Converse & If q, then p \\ \hline Contrapositive & If not q, then not p \\ \hline Inverse & If not p, then not q\\ \hline\end{tabular}[/Tex]
Note : It is interesting to note that the truth value of the conditional statement [Tex]p\rightarrow q
[/Tex]is the same as it’s contrapositive, and the truth value of the Converse of [Tex]p\rightarrow q
[/Tex]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,
- [Tex]p\rightarrow q \equiv \neg q\rightarrow \neg p
[/Tex]
- [Tex]q\rightarrow p \equiv \neg p\rightarrow \neg q
[/Tex]
[Tex]\begin{tabular}{ ||c||c||c||c||c||c||c||c|| } \hline p & q & \neg p & \neg q & p\rightarrow q & \neg q\rightarrow \neg p & q\rightarrow p & \neg p\rightarrow \neg q \\ \hline T & T & F & F & T & T & T & T \\ \hline T & F & F & T & F & F & T & T \\ \hline F & T & T & F & T & T & F & F \\ \hline F & F & T & T & T & T & T & T \\ \hline\end{tabular}[/Tex]
Example :
Implication : If today is Friday, then it is raining. The given proposition is of the form [Tex]p\rightarrow q
[/Tex], where [Tex]p
[/Tex]is “Today is Friday” and [Tex]q
[/Tex]is “It is raining today”. Contrapositive, Converse, and Inverse of the given proposition respectively are-
- Converse: If it is raining, then today is Friday
or
if q -> p is converse of p-> q
- Contrapositive: If it is not raining, then today is not Friday
or
if ~q -> ~p is contrapositive of p-> q
- Inverse: If today is not Friday, then it is not raining
or
if ~p -> ~q is inverse of p-> q
Implicit Use of Biconditionals
The last article, part one of this topic, ended with a discussion of bi-conditionals, what it is, and their truth table. In Natural Language bi-conditionals are not always explicit. In particular, the if construction (if and only if) is rarely used in the common language. Instead, bi-conditionals are often expressed using “if, then” or an “only if” construction. The other part of the “if and only if” is implicit, i.e. the converse is implied but not stated. For example consider the following statement, “If you complete your homework, then you can go out and play”. What is really meant is “You can go out and play if and only if you complete your homework”. This statement is logically equivalent to two statements, “If you complete your homework, then you can go out and play” and “You can go out and play only if you complete your homework”. Because of this imprecision in Natural Language, an assumption needs to be made whether a conditional statement in natural language includes its converse or not.
Precedence Order of Logical Connectives
Logical connectives are used to construct compound propositions 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-
[Tex]\begin{tabular}{ ||c||c|| } \hline Operator & Precedence \\ \hline \hline \neg & 1 \\ \hline \wedge & 2 \\ \vee & 3 \\ \hline \rightarrow & 4 \\ \leftrightarrow & 5 \\ \hline\end{tabular}[/Tex]
Here, higher the number lower the precedence.
Translating English Sentences
As mentioned above in this article, Natural Languages such as English are ambiguous i.e. a statement may have multiple interpretations. Therefore it is important to convert these sentences into mathematical expressions involving propositional variables and logical connectives. The above process of conversion may take certain reasonable assumptions about the intended meaning of the sentence. Once the sentences are translated into logical expressions they can be analyzed further to determine their truth values. Rules of Inference can then further be used to reason about the expressions.
Example : “You can access the Internet from campus only if you are a computer science major or you are not a freshman.” The above statement could be considered as a single proposition but it would be more useful to break it down into simpler propositions. That would make it easier to analyze its meaning and to reason with it. The above sentence could be broken down into three propositions,
[Tex]p[/Tex] - "You can access the Internet from campus."[Tex]q[/Tex] - "You are a computer science major."[Tex]r[/Tex] - "You are a freshman."
Using logical connectives we can join the above-mentioned propositions to get a logical expression of the given statement. “only if” is one way to express a conditional statement, (as discussed in Part 1 of this topic in the previous Article), Therefore the logical expression would be –
[Tex]p\rightarrow (q\vee \neg r)[/Tex]
Conclusion
In conclusion, this article describes about various laws of algebra of propositions, including Idempotent, Associative, Distributive, Commutative, Identity, De Morgan’s, and more. It also covers special conditional statements and the precedence order of logical connectives. Additionally, it emphasizes the importance of translating English sentences into mathematical expressions.
GATE CS Corner Questions
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.
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