Logical Connectives and Quantifiers: In any language, a statement is a sentence that you formally say or write that gives some information. However, in Mathematics, a sentence is called a statement if it is either true or false but not both. A statement may be simple or compound.

While a simple statement cannot be broken down into two or more statements, compound statements are formed by combining two or more statements using special words called connectives. Another important fragment of the construction of mathematical statements is quantifiers. The meaning attached to these special words plays an important role in validating the statements.

Definition of Logical Connectives

We know that a compound statement is a statement that is made up of two or more simple statements. In this case, each statement is called a component statement, and they are connected using special words called connectives.

For example, “The roses are red, or the ribbons are blue” is a compound statement. Similarly, “All squares are rectangles, and all rectangles are parallelograms”. Here, the first compound statement combines two simple statements using a connective or, also known as disjunction.
Whereas in the second statement, the connective used for combining the two simple statements is and, also known as a conjunction. The other important connectives that are used frequently are “not” or negation, “if-then” or conditional and “if and only if” or biconditional.

Basic Logical Connectives and their Validity

Simple statements are usually represented using small case alphabets like \(p,\,q,\) or \(r.\) The simple statement “All squares are rectangles” can be represented as \(p:\)

All squares are rectangles.

Conjunction

If two simple statements, \(p\) and \(q,\) are connected by the word “and,” then the resultant compound statement “\(p\) and \(q\) ” is called a conjunction of \(p\) and \(q.\)
Symbolically, conjunction is represented by ” \(\Lambda \)”. That is, the conjunction of simple statements \(p\) and \(q\) can be represented as

Validity

A conjunction is true only if both the component statements are true Logical connective example:
If \(p:\) All squares are rectangles
\(q:\) All rectangles are parallelograms
then, the conjunction of \(p\) and \(q\) can be represented as
\(p\Lambda q:\)All squares are rectangles, and all rectangles are parallelograms
Here, since \(p\) is true as well as \(q,\) the conjunction is true.

Disjunction

If two simple statements \(p\) and \(q,\) are connected by the word “or”, then the resultant compound statement “\(p\) or \(q\) ” is called a disjunction of \(p\) and \(q.\)
Symbolically, a disjunction is represented by “\( \vee \)”. That is, the disjunction of simple statements \(p\) and \(q\) can be represented as:

Validity

A disjunction is true if at least one of the component statements is true.
For example:
If \(p:\) The measure of the \(\angle A\) is greater than that of \(\angle B.\)
\(q:\)The measures of the angles \(\angle A\) and \(\angle B\) are equal.
Then, the disjunction of \(p\) and \(q\) can be represented as,
\(p \vee q:\)The measure of the \(\angle A\) is greater than that of \(\angle B,\) or they are equal.
This disjunction is true if the measure of \(\angle A\) is greater than or equal to that of \(\angle B.\)

Negation

The denial or a logical complement of a statement is called its negation. Though negation is not used to form a compound statement, it is considered a connective as it modifies the existing sentence. In general, the negation of a statement is formed by adding the connective “not” at the required place to mean the opposite of the given sentence. As a result, if a statement is true, then its negation is false. Other connectives used for negation are “it is false that” or “it is not the case that”.

If a statement is represented by \(p,\) the negation of \(p\) is represented as: For example, if \(p:\) The measures of vertically opposite angles are equal.
Then,
\( \sim p:\)The measures of vertically opposite angles are not equal.
Here, \(p\) is a true statement and \( \sim p\) is a false statement.

Negation of Compound Statements

Negation of a Conjunction

The negation of conjunction \(p \wedge q\) is the disjunction of the negation of \(p\) and negation of \(q.\)
That is, \( \sim \left( {p \wedge q} \right) = \, \sim p\, \vee \sim q.\)
Example: The negation of the compound statement “roses are red, and the sky is blue” is “roses are not red, or the sky is not blue”.

Negation of a Disjunction

The negation of disjunction \(p \vee q\) is the conjunction of the negation of \(p\) and negation of \(q.\)
That is, \( \sim \left( {p \vee q} \right) = \, \sim p\, \wedge \sim q.\)
Example: The compound sentence “\(17\) is a prime number or the sum of \(4\) and \(5\) is \(9\)” can be negated as “\(17\) is not a prime number and the sum of \(4\) and \(5\) is not \(9\)”.

Negation of a Negation

Negation of negation of a statement is the statement we started with. Symbolically, \( \sim \left( { \sim p} \right) = p.\)
Example: The negation of the statement “the base angles of an isosceles triangle are congruent” is “the base angles of an isosceles triangle are not congruent”. Now, the negation of the negation would be “It is not true that the base angles of an isosceles triangle are not congruent” or, in other words, “the base angles of an isosceles triangle are congruent”. That is, the negation of a statement is the statement itself.

Conditional Statement

A conditional statement is a statement in which the occurrence of a condition implies the occurrence of its result. For example, if I eat breakfast, then I feel good all day! Here, the condition of eating breakfast implies the result of feeling good all day.

The condition is also called the hypothesis. The result that the hypothesis implies is called the conclusion. The hypothesis and the conclusion are connected using the connectives if and then.

If \(p\) is a simple statement representing the hypothesis and \(q\) represents the conclusion, then the conditional statement can be represented as if \(p\) then \(q.\) We use an arrow starting from \(p\) directing to \(q\) to represent this using symbols.

Validity

In a conditional statement, the hypothesis is a sufficient condition for the conclusion. That is, the conclusion must be true if the hypothesis is true. So, a conditional statement is true if the hypothesis is true.

For example, if
\(p:\)A quadrilateral is a square.
\(q:\) It is a square.
Then,
\(p \Rightarrow q:\) If a quadrilateral is a square, then it is a rhombus.
This is a true statement as all the squares are rhombuses.

Contrapositive of a Conditional Statement

The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion and reversing the direction. That is, the contrapositive of the conditional statement \(p \Rightarrow q\) is \( \sim q \Rightarrow \sim p.\) In a conditional statement, the hypothesis is a sufficient condition for the conclusion. So, the negation of the conclusion automatically implies the negation of the hypothesis. Thus, a conditional statement and its contrapositive are logically equivalent statements!

For example, the contrapositive of the conditional statement “If a quadrilateral is a square, then it is a rhombus” would be “If a quadrilateral is not a rhombus, then it is not a square”. Since all squares are rhombuses, a quadrilateral not being a rhombus rightly implies it is not a square. Therefore, the conditional statement, as well as the contrapositive, are true!

Converse of a Conditional Statement

The converse of a conditional statement is obtained by interchanging the hypothesis and conclusion.
That is the converse of \(p \Rightarrow q\) is \(q \Rightarrow p.\)

Looking at the same example, “If a quadrilateral is a square, then it is a rhombus”, the converse of the statement would be, “If a quadrilateral is a rhombus, then it is a square”, which is false. Any quadrilateral with four congruent sides is a rhombus, but for a rhombus to be a square, it is necessary for all the four angles to be right! That is, all rhombuses are not squares! That is, the converse of a true statement need not be true.

Biconditional Statement

A biconditional statement combines a conditional statement and its converse using the connective if and only if. If \(p\) and \(q\) are two simple statements, then the biconditional statement \(p\) if and only if \(q\) is represented symbolically using a double-sided arrow as

Validity

A biconditional statement is true if and only both conditional statements and their converse are true.
For example, consider the following simple statements:
\(p:\)Four sides of a rectangle are equal.
\(q:\)It is a square.
Then, the conditional statement and its converse are:
\(p \Rightarrow q:\) If four sides of a rectangle are equal, then it is a square.
\(q \Rightarrow p:\) If a rectangle is a square, then its four sides are equal.
Here, both the conditional statement and its converse are true statements.
Now, the biconditional statement can be written as:
\(p \Leftrightarrow q:\) A rectangle is a square if and only if its four sides are equal.

Quantifiers

Quantifiers are another type of phrase or a special word used in mathematical statements. They refer to quantities that the number of elements in the domain that satisfy the particular simple statement. The most commonly used logical quantifiers examples are “there exist”, “for all”, “for some”, and “for every”.

There are mainly two types of quantifiers:

Universal Quantifier

These expressions are used to assert that the mentioned statement is true for all the domain members. The quantifier phrases used in this case are for all, for every, for each, for any etc. Symbolically this quantifier is expressed as

Example:
\(p:\) For every prime number \(x,\sqrt x \) is irrational
In symbols \( – \forall x \in pZ,\,\sqrt x \in Q\) where \(pZ\) is the set of prime numbers and \(Q\) is the set of rational numbers.
Or
\(q:\) For any real number \(x,\) if \(x = y,\) then \(3x + m = 3y + m\)
Using symbols \( – \forall x \in R,\,x = y \Rightarrow 3x + m = 3y + m\)

Existential Quantifier

These types of quantifiers refer to the existence of elements in the domain with particular properties. The phrases used exist, for some, for at least one, there is a, etc. In symbols, they are represented by

Example:
\(p:\) There exists a real number that is equal to its square
In symbols \( – \exists x \in R,\,{x^2} = x\)
Or
\(q:\) In the set of triangles, there is at least one isosceles triangle
In symbols \( – \exists T \in \) Set of Triangles, \(T\) is isosceles

Solved Examples – Logical Connectives and Quantifiers

Q.1. Translate the statements into the symbolic form:
(i) \(x\) and \(y\) are even integers
(ii) A number is either divisible by \(2\) or \(3\)
Ans:
(i) \(p:x\) is an even integer
\(q:y\) is an even integer and
\(p \wedge q:x\) and \(y\) are even integers.
(ii) \(p:\) A number is divisible by \(2\)
\(q:\) A number is divisible by \(3\)
\(p \vee q:\) A number is either divisible by \(2\) or \(3.\)

Q.2. If \(p:100\) is divisible by \(4\) and \(q:100\) is divisible by \(5,\) check whether the conjunction, disjunction and negations of the statements are true.
Ans: The statements \(p\) and \(q\) are true because \(100\) is divisible by the numbers \(4\) and \(5.\)
Conjunction: Since both \(p\) and \(q\) are true, the conjunction \(p \wedge q\) is true.
Disjunction: Both \(p\) and \(q\) are true, and for a disjunction to be true, it is enough for at least one of the statements to be true. So, disjunction \(p \vee q\) is also true.
Negation: Negation of a true statement is false. So, both \( \sim p\) and \( \sim q\) are false.

Q.3. Rewrite each of the following statements in the form of conditional statements
Ans: The square of a prime number is not a prime number
The sentence can be rewritten as “If \(x\) is a prime number, then \({x^2}\) is not a prime number.”
So, the component statements can be written as is a \(p:x\) prime number and \(q:{x^2}\) is not a prime number and the conditional statement \(p \Rightarrow q:\) If \(x\) is a prime number, then \({x^2}\) is not a prime number.

Q.4. Form the biconditional statement \(p \Leftrightarrow q,\) where
\(p:\) The unit digit of an integer is zero
\(q:\) It is divisible by 5
Ans: The conditional statements are:
\(p \Rightarrow q:\) If the unit digit of an integer is \(0\) then it is divisible by \(5\)
\(q \Rightarrow p:\) If an integer is divisible by \(5\) then the unit digit of the integer is zero
The biconditional statement is:
\(p \Leftrightarrow q:\) The unit digit of an integer is zero if and only if it is divisible by \(5\)
A biconditional statement is true if both the conditional statements are true. Here, though \(p \Rightarrow q\) is true, its converse \(q \Rightarrow p\) is false and therefore \(p \Leftrightarrow q\) is false.

Q.5. Identify the quantifiers and write the negation of the following statement
(i) There exists a real number which is not a rational number
(ii) For every natural number \(x,\,x + 1\) is also a natural number
Ans: (i) Here, the quantifier phrase is “there exist”.
The negation of the statement is “All real numbers are rational numbers”.
(ii) The quantifier phrase used is “for every”.
The negation is “There exist a natural number \(x\) such that \(x + 1\) is not a natural number.”

Summary

The article starts with defining logical connectives and moves ahead to list all the five logical connectives such as conjunction, disjunction, negation, conditional and biconditional. The article also discusses the examples and the usages of each connective in detail. It also explains the negations of compound statements obtained by using these connectives as an added topic.

Further quantifiers are defined, and the two types of the same are illustrated with ample examples for strengthening the concepts. Solved examples are provided to reinforce the ideas discussed in the whole content.

Frequently Asked Questions (FAQs)

Q.1. What are the different logical connectives?
Ans: The common phrases used for these are listed below:

Connective	Phrases	Symbols
Conjunction	and	\( \wedge \)
Disjunction	or	\( \vee \)
Negation	not	\( \sim \)
Conditional	If-then	\( \to \) or  \( \Rightarrow \)
Biconditional	If and only if	\( \leftrightarrow \)  or \( \Leftrightarrow \)

Q.2. What are logical connectives and universal quantifiers?
Ans: Special words or phrases that are used to combine or modify simple statements to form new or compound statements are called connectives. The main logical connectives are conjunction, disjunction, negation, conditional and biconditional. Quantifiers are another type of phrase or a special word used in mathematical statements. Universal quantifiers are expressions used to assert that the mentioned statement is true for all the domain members. The quantifier phrases used in this case are for all, for every, for each, for any etc. Symbolically this quantifier is expressed as \(\forall .\)

Q.3. How do you express a statement using quantifiers?
Ans: Universal quantifiers are expressed using the symbol \(\forall \) and the existential quantifiers by \(\exists .\)
For example, “for every integer \(n,\,{n^3} – n\) is an even number” uses a universal quantifier. Using symbols, this can be written as \(\forall n \in I,\,{n^3} – n\) is even.
The statement “there exists an even prime other than \(2\)” is a false statement that uses an existential quantifier. This can be written as \(\exists x \in N,\,x\) is a prime and \(x\) is even.

Q.4. What are the \(2\) types of quantifiers?
Ans: The two types of quantifiers are universal quantifiers and existential quantifiers. Universal quantifiers are used to assert that the mentioned statement is true for all the members of the domain, whereas existential quantifiers refer to the existence of elements in the domain with particular properties. Universal quantifiers are expressed using the symbol \(\forall \) and the existential quantifiers by \(\exists .\)

Q.5. How do you identify quantifiers?
Ans: Quantifiers are identified using the special words used in the statements. In universal quantifiers, commonly used phrases are for all, for every, for each, for any. In contrast, in existential quantifiers, phrases used are: there exist, for some, and for at least one.

We hope this detailed article on logical connectives and quantifiers helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!