Implications: A logical statement is a meaningful sentence that cannot be true and false simultaneously. A conditional statement has a set condition according to which if a particular statement is true, then the other specified statement is true too. We can also reframe this, as the truth of a specific statement implies the validity of some other specified statement.

Thus, we can say that conditional statements are also called Implications. For example, “If we leave early, we can reach on time” is a conditional statement of the form “If… then…”. We can also write this as “leaving early implies we reach on time.” which is an implication. Let us now learn about the Implications in this article.

Implications: Definition and Properties

An implication is a compound statement, which means it has more than one logical statement. It is a conditional statement of the form “If $A$, then $B$.” The mathematical notation of an implication is $\to$ or $\Rightarrow$.

“If $A$, then $B$.” is written symbolically as $A\to B$ or $A\Rightarrow B$.

In any implication,

• In the above example, the first statement, $A$, is called the antecedent.
• The implied statement or the second statement, $B$ in the above example, is called the consequent.

The implication $A\to B$ is false only when $A$ is true, and $B$ is false. When this happens, we can say that $A$ does not imply $B$. In simpler words, the implication is false.

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Examples:

• “If it rains today, then we will play football in the mud.”
• “If we paint the door with a varnish, then it will last longer.”
• “Being over $18$ is a sufficient condition to be eligible to vote in our country.”

Ways to Express Implications

Some of the ways to write implications are:

• If $A$, then $B$.
• If $A$, $B$.
• $A$ if $B$.
• A necessary condition for $A$ is $B$.
• $A$ implies $B$.
• $B$ only if $A$.
• $A$ is necessary for $B$.
• $B$ follows $A$.

Look out for these, and you can easily identify an implication.

Truth Table of a Logical Implication

A truth table covers all the possibilities of the truth values of the individual statements in an implication. Hence, it is extensively used in logic and discrete Mathematics to find the true value of compound statements.

$A$	$B$	$A\to B$
$T$	$T$	$T$
$T$	$F$	$F$
$F$	$T$	$T$
$F$	$F$	$T$

Note that the implication $A\to B$ is false only when $A$ is true, and $B$ is false, and it is true in all other cases.

Types of Implications

Consider an implication we have seen before $A\to B$.

Converse of $A\to B$ is $B\to A$

Inverse of $A\to B$ is $\sim A\to \sim B$

Contrapositive of $A\to B$ is $\sim B\to \sim A$

Here,

$\sim A$ means negation of $A$.

$\sim B$ means negation of $B$.

Where Can We Use Implications?

It is important to note that implications regularly find application in logical arguments of mathematical proofs.

Example:

If a perpendicular line is drawn from the centre of a circle to a chord, (then) it will bisect the chord.

Notice that using ‘then’ is not mandatory.

• Antecedent – “a perpendicular drawn from the centre of a circle to a chord”
• Consequent – “it will bisect the chord”

If we interchange the antecedent and the consequent of an implication, the resulting statement is the converse of the given implication.

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Bi-conditional Statements

As the name suggests, it has two conditions and is also called a double implication. It is of the form “$A$ if and only if $B$.”

The mathematical notation of a double implication is $\leftrightarrow$ or $\iff$.

“$A$ if and only if $B$.” is written symbolically as $A\to B$ or $A\iff B$.

• The implication $A\to B$ is false when $A$ and $B$ have different truth values.
• It is a combination of an implication and the converse of an implication.

Example:

$A$: A number is divisible by $10$.

$B$: The digit in its unit’s place is $0$.

We can say $A\to B$.

If a number is divisible by $10$, the digit in its unit’s place is $0$.

Hence, we can also say $B\to A$.

If a digit in the unit’s place is $0$, it is divisible by $10$.

$\therefore \mathbf{A}\leftrightarrow \mathbf{B}\equiv (\mathbf{A}\to \mathbf{B})\wedge (\mathbf{B}\to \mathbf{A})$

$\wedge$ is the logical AND operator

Different Ways to Express Double Implications

Some ways to write double implications are:

• $A$ if and only if $B$.
• $B$ if and only if $A$.
• $A$ is a necessary and sufficient condition for $B$.

“if and only if” can also be written as “iff”

• $A$ iff $B$.

Truth Table of a Logical Double Implication

The truth table for a logical double implication is shown below:

$A$	$B$	$A\leftrightarrow B$
$T$	$T$	$T$
$T$	$F$	$F$
$F$	$T$	$F$
$F$	$F$	$T$

Note that the implication $A\leftrightarrow B$ is false when $A$ and $B$ have different truth values.

Where Can We Use Double Implications?

Double implications are helpful in the case of mathematical proofs where the converse of the proof is also true.

Solved Examples – Implications

Let us look at some of the solved examples about Implications:

Q.1. Express the following statements in the symbolic form.
a. If the bus drops us off on time, then we can catch the connecting flight.
b. Price increases if and only if demand falls.
Ans:
a. Let $x$ and $y$ be two statements such that,
● $x$ denotes “the bus drops us off on time”
● $y$ denotes “we can catch the connecting flight”
Symbolic form of the given compound statement is $x\to y$.

b. Let $r$ and $s$ be two statements such that,
● $r$ denotes, “price increases”
● $s$ denotes “demand falls”
Symbolic form of the given compound statement is $r\leftrightarrow s$.

Q.2. Rewrite each of the following statements in “if-then” form.
a. The fact that you can run fast implies that you must have been practising daily.
b. Two triangles are similar if their corresponding sides are in proportion.
Ans:
a. If you can run fast, then you must have been practicing daily.
b. If the corresponding sides of a triangle are in proportion, then the two triangles are similar.

Q.3. In the given question, $p,q,r$ and $s$ are different statements or propositions. It given that the statements $p$ and $q$ are true and $r$ and $s$ are false. Using this information, find the true value of each of the following statements.
a. $\left[\left(p\vee s\right)\to r\right]\vee \sim \left[\sim \left(p\to q\right)\vee s\right]$
b. $\left(q\to r\right)\vee \left(\sim p\to \sim q\right)$
Ans:
a. It is given that statements $p$ and $q$ are true, and statements $r$ and $s$ are false.
Substituting these truth values in the compound statement, we get,
$[(p\vee s)\to r]\vee \sim [\sim (p\to q)\vee s]$
$\equiv [(T\vee F)\to F]\vee \sim [\sim (T\to T)\vee F]$
$\equiv (T\to F)\vee \sim (\sim T\vee F)\dots .(\because T\vee F\equiv T\mathrm{a}\mathrm{n}\mathrm{d}T\to T\equiv T)$
$\equiv \mathrm{F}V\sim (F\vee F)\dots .(\because T\to F\equiv F\mathrm{a}\mathrm{n}\mathrm{d}\sim T\equiv F)$
$\equiv F\vee \sim F\dots .(\because F\vee F\equiv F)$
$\equiv F\vee T\dots .(\because \sim F\equiv T)$
$\equiv T\dots .(\because F\vee T\equiv T)$

b. Substituting the given truth values in the compound statement, we get,
$(q\to r)\vee (\sim p\to \sim q)$
$\equiv (T\to F)\vee (\sim T\to \sim T)$
$\equiv F\vee (F\to F)\dots .(\because T\to F\equiv F\mathrm{a}\mathrm{n}\mathrm{d}\sim T\equiv F)$
$\equiv F\vee T\dots (\because F\to F\equiv T)$
$\equiv T\dots .(\because F\vee T\equiv T)$

Q.4. Create truth tables for the compound statements.
a. $[(p\to q)\vee p]\to p$
b. $(\sim p\vee \sim q)\leftrightarrow \sim (p\wedge q)$
Ans:
a. There will be $2^n+1$ number of rows in the truth table, where $n$ is the number of different statements.
Since there are two different statements $p$ and $q$ in this compound statement, there will be $2^2+1=5$ rows in the truth table.

$p$	$q$	$p\to q$	$(p\to q)\vee p$	$[(p\to q)\vee p]\to p$
$T$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$T$	$T$
$F$	$T$	$T$	$T$	$F$
$F$	$F$	$T$	$T$	$F$

b. There will be $2^n+1$ number of rows in the truth table, where $n$ is the number of different statements.
Since there are two different statements $p$ and $q$ in this compound statement, there will be $2^2+1=5$ rows in the truth table.

$p$	$q$	$\sim p$	$\sim q$	$\sim p\vee \sim q$	$p\wedge q$	$\sim (p\wedge q)$	$(\sim p\vee \sim q)\leftrightarrow \sim (p\wedge q)$
$T$	$T$	$F$	$F$	$F$	$T$	$F$	$T$
$T$	$F$	$F$	$T$	$T$	$F$	$T$	$T$
$F$	$T$	$T$	$F$	$T$	$F$	$T$	$T$
$F$	$F$	$T$	$T$	$T$	$F$	$T$	$T$

Q.5. Make truth tables for each of the following compound statements. Then, state whether each of these is a tautology, a contradiction or a contingency.
a. $(p\to q)\wedge (p\wedge \sim q)$
b. $(p\wedge q)\to q$
Ans: Tautology – A compound statement that always has truth value $‘T^{\prime }$, irrespective of the truth values of its component statements
Contradiction – A compound statement that always has truth value $‘F^{\prime }$, irrespective of the truth values of its component statements
Contingency – A compound statement which is neither a tautology nor a contradiction

a. There will be $2^n+1$ number of rows in the truth table, where $n$ is the number of different statements.
Since there are two different statements $p$ and $q$ in this compound statement, there will be $2^2+1=5$ rows in the truth table.

$p$	$q$	$p\to q$	$\sim q$	$p\wedge \sim q$	$(p\to q)\wedge (p\wedge \sim q)$
$T$	$T$	$T$	$F$	$F$	$F$
$T$	$F$	$F$	$T$	$T$	$F$
$F$	$T$	$T$	$F$	$F$	$F$
$F$	$F$	$T$	$T$	$F$	$F$

All truth values are $‘F^{\prime }$ which implies that this statement is a contradiction.

b. There will be $2^n+1$ number of rows in the truth table, where $n$ is the number of different statements.
Since there are two different statements $p$ and $q$ in this compound statement, there will be $2^2+1=5$ rows in the truth table.

$p$	$q$	$p\wedge q$	$(p\wedge q)\to q$
$T$	$T$	$T$	$T$
$T$	$F$	$F$	$T$
$F$	$T$	$F$	$T$
$F$	$F$	$F$	$T$

All truth values are $‘T^{\prime }$, which implies that this statement is a tautology.

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Summary

An implication is a conditional compound statement of the form “If $A$, then $B$.” denoted as $A\to B$. It is also expressed as $A$ implies $B$. The implication $A\to B$ is false only when $A$ is true, and $B$ is false. It is true in all other cases. A double implication is a compound statement of the form “$A$ if and only if $B$.” denoted as $A\leftrightarrow B$. The implication $A\leftrightarrow B$ is false, when $A$ and $B$ have different truth values.

Implications are used in logical arguments like mathematical proofs. Double implications are used in the case of mathematical proofs where the converse of the proof is also true. The other types of statements discussed are converse, inverse, and contrapositive of implications.

Frequently Asked Questions (FAQs)

Q.1. How do you write an implication in Maths?
Ans: Some of the ways to write implications are:

• If $A$, then $B$.
• If $A,B$.
• $A$ if $B$.
• A necessary condition for $A$ is $B$.
• $A$ implies $B$.
• $B$ only if $A$.
• $A$ is necessary for $B$.
• $B$ follows $A$.

Q.2. What is an example of an implication?
Ans: A good monsoon implies an increase in water reserves is an example of an implication. Here “increase in water reserves” follows “good monsoon”.

Q.3. What do you mean by implications?
Ans: A conditional statement has a set condition according to which if a particular statement is true, then the other specified statement is also true. This is called an implication.

Q.4. What is the symbol of implication?
Ans: The mathematical notation of an implication is $\to$ or $\Rightarrow$.

Q.5. What is a logical implication in Discrete Maths?
Ans: Implication is a conditional compound statement. It is common in the form “If $A$, then $B$.” and “A necessary condition for $A$ is $B$.”

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