Propositional Logic (statements)

To understand the meaning and have a clear picture about propositional logic we first have to know what a logic is. Logic is the science of reasoning. It helps us knowing the reason behind different mathematical statements and finally we would be able to prove or disprove any statement precisely, e.g., if I say “You are a student of class IX” then there must be reason behind the statement. Now, I’m supposed to present a logic which can be like “There are some textbooks in your bag those are for class IX”.

The purpose of logic to construct valid arguments (or proofs). Once proved a mathematical statement as true then we call it a theorem, and this is the basis of the whole mathematics.

A proposition (or a statement) is a declarative sentence (a sentence that is declaring a fact or stating an argument) which can be either true or false but cannot be both, e.g.,

• Dispur is the capital of Assam
• Jaipur is the capital of Bhutan
• \(1+1=2\)
• It's a rainy day today.
• Columbus discovered America
• Washington DC is in India.
• India won the match by \(5\) runs.
• \(5\) is an even number.
• The only odd prime number is \(2\).
• \(5+7=10\)
• \(-2\) is a natural number.
• The sun rises in the east.

On the other hand, the following sentences are not statements-

• What is the time now?
• Where do you live?
• Shut the door.
• I request you to join my party.
• Fetch my umbrella.
• Do your homework properly.
• I request you to look after my child for a day.
• God bless you.

Propositional Logic

It is the area of logic that studies the ways of joining and/or modifying statements to form more complicated statements and it also studies the logical relationships and properties derived from these combined/altered statements. When a statement is formed by combining two or more simple statements then we call it as a composite or compound statement, and its component are called sub-statement. The terms like 'and' 'or' 'nor' 'if-then' 'if and only if' etc. which connects the sub-statements are called connectives, e.g.,
Statement₁: Sujan is good in playing badminton.
Statement₂: Sujan is good in playing badminton and this time he is representing his school at state level.

Here, Statement₁ is a sub-statement and Statement₂ is a compound statement. Also, we can see that there are two sub-statements in Statement₂ and both are connected by the connective 'and'.

Propositional Variables

The variables that are used to represent statements are called propositional variables, e.g., Sujan is good in playing badminton and this time he is representing his school at state level. If we break the statement in the two of its component statements as-
\(p\) = Sujan is good in playing badminton
\(q\) = this time he is representing his school at state level
then \(p\) and \(q\) are called the propositional variables.

The various types of statements are discussed as follows-

Negation

If \(p\) be a statement then \(\sim{p}\) is called the negation of \(p\) which simply states that “it is not the case \(p\)”, e.g., if the statement \(p\) means ‘Mercury is heavier than Gold’ then \(\sim{p}\) will mean ‘it is not the case that Mercury is heavier than Gold’ or ‘it is false that Mercury is heavier than Gold’ or ‘it is not true that Mercury is heavier than Gold’.

Conjuction

If \(p\) and \(q\) be two statements then conjunction of \(p\) and \(q\) is denoted by \(p \wedge{q}\) . If both \(p\) and \(q\) are true only then the compound statement \(p \wedge{q}\) is true. In this case, \(p \wedge{q}\) is true only when both \(p\) and \(q\) are true, otherwise false.

Examples:

• Squares are rectangles and rectangles have four sides
• He is cleaver but lazy.

Disjunction

If \(p\) and \(q\) be two statements then disjunction of \(p\) and \(q\) is denoted by \(p \vee{q}\). In this case, the compound statement \(p \vee{q}\) is always true except when both \(p\) and \(q\) are false.

Examples:

• Some quadrilaterals are parallelograms or quadrilaterals have 11 sides.
• All numbers are integers or squares are rectangles.

Implications or conditional

If \(p\) and \(q\) are two statements, then the statements ‘if \(p\) then \(q\)’ denoted by \(p \to{q}\) is called implication or conditional statement. Here, \(p\) is called hypothesis (or premise) and \(q\) is called conclusion (or consequence). In this case, the compound statement \(p \to{q}\) is always true, except when \(p\) is true and \(q\) is false.

One of the examples is verified as follows-

1. If you work hard for your exam then you will be succeeded.
• If you work hard for your exam then you will be succeeded
  Here if, \(p\) = you work hard for your exam = T
  and \(q\) = you will be succeeded = T
  then, \(p \to{q}\) = T
  
• you tried hard for your exam but you failed
  Here if, \(p\) = you tried hard for your exam = T
  and \(q\) = you failed = F
  then, \(p \to{q}\) = F
  
• you haven’t tried hard for your exam but you succeeded
  Here if, \(p\) = you haven’t tried hard for your exam = F
  and \(q\) = you succeeded = T
  then, \(p \to{q}\) = T
  
• you haven’t tried hard for your exam and you failed
  Here if, \(p\) = you work hard for your exam = F
  and \(q\) = you will be succeeded = F
  then, \(p \to{q}\) = T
  
2. If you get 80% marks on your exam then you will get a laptop.
3. If you have connections with seniors then you will get promoted.

Biconditional

If \(p\) and \(q\) are two statements then the compound statement ‘\(p\) if and only if \(q\)’ denoted by \(p \leftrightarrow q\) is called biconditional statement. In this case, \(p \leftrightarrow q\) is true only when the truth values of \(p\) and \(q\) are the same. Here, ‘\(p\) if and only if \(q\)’ is composed of two statements- ‘\(p\) if \(q\)’ and ‘\(p\) only if \(q\)’ and its formation is as follows-
‘\(p\) only if \(q\)’= if \(p\) then \(q\) and ‘\(p\) if \(q\)’ = if \(p\) then \(q\)
⇒ \((p \to{q}) \wedge{(q \to{p}) }= p \leftrightarrow q\)

Representation

• \(p\) is necessary and sufficient for \(q\) and vice-versa.
• If \(p\) then \(q\) and the converse.
• \(p\) if and only if \(q\).

Examples:

• You will get promoted if and only if you have connections with your seniors.
• I will stay at home if and only if I’m sick.

Precedence of the logical operators

It helps us to decide which operator will get evaluated first in a complicated looking compound statement and the order is as follows: negation, conjunction, disjunction, conditional, biconditional.

• Sometimes whatever the values of the components statements \(p\) and \(q\), the value of the given statement is always true(T), such a statement is called a formula or logically valid or a tantology.
• Two statements are called equivalent (or logically equivalent) if one is true then the other is also true and if one is false then the other is also false, e.g., \(4x = 8\) and \(8x = 16\) are equivalent, because for \(x = 2\) both are true but for any other value of \(x\) both are false.

Converse, Inverse and Contrapositive Statements

Let \(p\) and \(q\) be two statements, then

• Implication or conditional statement: \(p \to{q}\)
• Converse statement: \(q \to{p}\)
• Inverse statement: \(\ \sim{p} \to{\sim{q}} \)
• Contrapositive statement: \(\ \sim{q} \to{\sim{p}} \)

Examples: If it rains today then I will stay at home.

• Converse statement: If I stay at home then it will rain today.
• Inverse statement: If it doesn’t rain today then I’ll not stay at home.
• Contrapositive statement: If I don’t stay at home then it will not rain today.

Relations in a nutshell with a truth table:

• Both the conditional and contrapositive statements are equivalent.
• Both the converse and inverse statements are equivalent.
• Neither converse statement nor inverse statement is equivalent to conditional statement.

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