Logic Connectivity in Mathematical Reasoning

In mathematics, the study of logic is fundamental to understanding and constructing valid arguments. A significant aspect of logical reasoning involves the use of connectives, which are words or symbols that combine or alter simple statements to create more complex statements known as compound statements.

Logical Connectives

The words which combine or change simple statements to form new statements or compound statements are called Connectives. The basic connectives (logical) conjunction corresponds to the English word ‘and’, disjunction corresponds to the word ‘or’, and negation corresponds to the word ‘not’.

Negation of a Statement

or "It is false that".

The negation of a statement $p$ in symbolic form is written as " $p$" and read as "not $p$".
$p$ : New Delhi is the capital of India.
The negation of this statement is
$\sim$ $p$ : New Delhi is not the capital of India.
Or, $\quad \sim p$ : It is not the case that New Delhi is the capital of India.
Or, $\quad \sim p:$ It is false that New Delhi is the capital of India.

Conjunction

If two simple statements $p$ and $q$ are connected by the word 'and', then the resulting compound statement " $p$ and $q$ " is called a conjunction of $p$ and $q$ and is written in symbolic form as " $p \wedge q$ ".
For example,
$\mathrm{p}:$ Delhi is in India and $2+3=5$.
The statement can be broken into two component statements as
$q$: Delhi is in India.

$
r: 2+3=5
$

The compound statement with 'And' is true if all its component statements are true.
The component statement with 'And' is false if any of its component statements are false.

Note:
A statement with "And" is not always a compound statement.
For example,
$p$ : A mixture of alcohol and water can be separated by chemical methods
Here the word "And" refers to two things - alcohol and water.

Disjunction

If two simple statements $p$ and $q$ are connected by the word 'or', then the resulting compound statement " $p$ or $q$ " is called a disjunction of $p$ and $q$ and is written in symbolic form as " $p \vee q$ ".
$p$: Delhi is in India or $2+3=5$.
The statement can be broken into two component statements as
$q:$ Delhi is in India.

$
r: 2+3=5
$
The compound statement with 'or' is true if any of its component statements are true.
The component statement with 'or' is false if all of its component statements are false.
Types of OR statements
1. Inclusive OR: If $p$ and $q$ can simultaneously be true, then we say that 'p or q' has an inclusive OR.

Eg, 'Bangalore is in Karnataka or India'
Here both component statements 'Bangalore is in Karnataka' and ' 'Bangalore is in India' can be true simultaneously. Hence this compound statement has an inclusive OR.

2. Exclusive OR: If $p$ and $q$ cannot be true simultaneously, then we say that '$p$ or $q$ ' has an exclusive OR.

Eg, 'Bangalore is in Karnataka or Maharashtra'
Here both component statements 'Bangalore is in Karnataka' and ' 'Bangalore is in Maharashtra' cannot be true simultaneously. Hence this compound statement has an exclusive OR.

Conditional Statement
$r$: If you are born in some country, then you are a citizen of that country
$p$ : you are born in some country.
$q$: you are a citizen of that country.
Then the sentence "if $p$ then $q$" says that in the event if $p$ is true, then $q$ must be true.

The conditional statement $p \Rightarrow q$ can be expressed in several different ways. Some of the common expressions are
1. $p$ implies $q$
2. $p$ is a sufficient condition for $q$
3. $p$ only if $q$
4. $q$ is necessary condition for $p$.
5. $\sim q$ implies $\sim p$

The statement $p \rightarrow q$ is false only when $p$ is true and $q$ is false. In all other cases this statement is true.

The Biconditional Statement

If two statements $p$ and $q$ are connected by the connective "if and only if then the resulting compound statement " $p$ if and only if $q$ " is called a biconditional of $p$ and $q$ and is written in symbolic form as $p \leftrightarrow q$ or $p \Leftrightarrow q$.
ne segments are congruent if and only if they are of equal length'
It is a combination of two conditional statements, "if two line segments are congruent then they are of equal length" and "if two line segments are of equal length then they are congruent". Which means
$p \leftrightarrow q$ is same as ${ }^{\prime} p \rightarrow q$ AND $q \rightarrow p^{\prime}$
A biconditional is true if and only if both the statements $p \rightarrow q$ and $q \rightarrow p$ are true
Also a biconditional is true if $p$ and $q$ both are true or when both $p$ and $q$ are false.

Solved Examples Based on Logical Connectives:

Example 1: What does symbol " $\wedge$ " depicts?
1)OR
2)AND
3) NOT
4) IF

Solution
Conjunction -
Symbol " $\Lambda$ " is used to denote conjunction.
Symbol "^" depicts AND

Example 2: Which one is NOT an example of an AND conjunction?
1) $p: x+y=3$ and $x-y=1$
2) $q$: $x^2-4>0$ and $y^2-3<0$
3) $r$ : Sam opened the closet and took out clothes
4) $S$: Delhi in India and Mumbai is in Europe

Solution

'And' Conjunction -

Normally the conjunction 'and' is used between two statements which have some kind of relation but in logic, it can be used even if there is no relation between the statements.

"Sam opened the closet and took out clothes" is not an example of an AND conjucton as "and " is used in a different sense here.

Example 3: Find the entire truth set of $x^4-1=0$
1) $\{-1,1\}$
2) $\{1\}$
3) $\{i,-1,1,-i\}$
4) none of these

Solution
Logic Connectivity -
Note:

$
\begin{aligned}
& x^4-1=0 \\
& \left(x^2-1\right)\left(x^2+1\right)=0 \\
& \Rightarrow x= \pm 1
\end{aligned}
$
$
\text { or, } x^2=-1 \Rightarrow x= \pm \sqrt{-1} \Rightarrow x= \pm i
$

Example 4: How will you prove that $\sqrt{3}$ is irrational?
1) put $\sqrt{3}=p+i q$
2) put $\sqrt{3}=\sqrt{p}$
3) put $\sqrt{3}=\frac{p}{q}$
4) none of these

Solution
Logic Connectivity -
Truth Value of a Statement
As we know that a statement is either true or false. The truth or falsity of a statement is called truth value.
If the statement is true, then truth value is " T "
If the statement is false, then truth value is " $F$ "
It can be done by contradiction i.e assuming $\sqrt[4]{a}$

Example 5: Let $S$ be a non-empty subset of $R$. Consider the following statement:
$P$ : There is a rational number $x \in S$ such that $x>0$.

Which of the following statements is the negation of the statement $P$ ?
1) There is a rational number $x \in S$ such that $x \leq 0$.
2) There is no rational number $x \in S$ such that $x \leq 0$.
3) Every rational number $x \in S$ satisfies $x \leq 0$.
4) $x \in S$ and $x \leq 0 \Rightarrow x$ is not rational.

Solution
Logic Connectivity -
Negation of a Statement
The negation of a statement $p$ in symbolic form is written as " $\sim$ " and read as "not $p$".
$p$: New Delhi is the capital of India.
The negation of this statement is
$\sim$ p: New Delhi is not the capital of India.
Or,
$\sim \underline{p}$ : It is not the case that New Delhi is the capital of India.
Or,
$\sim$ p; It is false that New Delhi is the capital of India.

The truth value of negation of a statement is always opposite to the truth value of the original statement.

$\begin{array}{|c|c|}\hline \mathrm{\;\;\;\;\;}p\mathrm{\;\;\;\;\;}&\mathrm{\;\;\;}\sim p\mathrm{\;\;\;\;\;} \\ \hline \mathrm{T}& \mathrm{F} \\ \hline \mathrm{F}&\mathrm{T} \\ \hline\end{array}$

We write it up as $\sim p$
$P$ : There is rational numbers $x \epsilon S$ such that $x>0$. negation is every rational number $x \epsilon S$ satisfies $x \leq 0$ \{ Since negation of $x>0$ is $x \leq 0$.

Summary

Logical connectives are essential tools in mathematics and logic, allowing for the construction of complex statements from simpler ones. By understanding how conjunction, disjunction, negation, conditional, and biconditional connectives function, one can form precise logical arguments and engage in rigorous reasoning. These connectives not only help in mathematical proofs but also in everyday logical thinking, making them fundamental components of clear and effective communication.