Logic Connectivity - Practice Questions & MCQ

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

The denial of a statement is called the negation of the statement. The negation of a statement is generally formed by introducing the word “not” at some proper place in the statement or by prefixing the statement with “It is not the case that” 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
$\sim \mathrm{p}$ : New Delhi is not the capital of India.
Or, p p : It is not the case that New Delhi is the capital of India.
Or, p 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 $\vee \mathrm{q}$ ".

For example,
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

If $p$ and $q$ are any two statements, then the compound statement "if $p$ then $q$ " formed by joining $p$ and q by a connective 'if then' is called a conditional statement or an implication and is written in symbolic form as $p\to q$ or $p\Rightarrow q$.
$r$ : 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 $p\to q$ 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 $p\iff q$.

Eg, 'Two line 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

$p\leftrightarrow q\text{ is same as ' }p\to q\text{ AND }q\to p^{\prime }$
A biconditional is true if and only if both the statements $p\to q$ and $q\to p$ are true
Also a biconditional is true if $p$ and $q$ both are true or when both $p$ and $q$ are false.