Question

What is the meaning of $P\to Q$?

Answer 1

Hint: For solving this question first we will understand the meaning of “Implication” or “Conditional” statements. As we know that in mathematics we come across many statements of the form $\text{''if }P\text{ then }Q''$ and it can be represented as $P\to Q$. Such statements are called conditional statements. In this question, we will discuss such statements with the help of practical examples and then we will be able to answer the question comfortably.

Complete step-by-step solution -

Given:
We have to explain the meaning of $P\to Q$.
Now, for answering the question we should know “Implication” or “Conditional” statements.
IMPLICATION OR CONDITIONAL STATEMENTS:
Two statements connected by the connective phrase “if” then give rise to a compound statement which is known as an implication or conditional statement. For example: “If tea is hot, then serve it”, “If $x=2$, then ${{x}^{3}}=8$ ”, “ If ABCD is a square, then AB=BC” are implications.
Now, if $P$ and $Q$ are two statements, the compound statement $\text{''if }P\text{ then }Q''$ is called an implication or conditional statement. And if $P$ and $Q$ are two statements forming the implication $\text{''if }P\text{ then }Q''$ , then we denote this implication by “ $P\to Q$ ” or “ $P\Rightarrow Q$ ”.
Now, in the implication “ $P\to Q$ ”, $P$ is called the antecedent or hypothesis and $Q$ the consequent or conclusion. The truth value of an implication $P\to Q$ depends on the truth values of its antecedent $P$ and consequent $Q$ . For the truth table of $P\to Q$ there will be a total four cases. In the first case when $P$ is true (T) and $Q$ is also true (T) then, $P\to Q$ will be true (T). And in the second case when $P$ is false (F) and $Q$ is true (T) then, $P\to Q$ will be true (T). And in the third case when $P$ is true (T) and $Q$ is false (F) then, $P\to Q$ will be false (F). And in the fourth case when $P$ is false (F) and $Q$ is also false (F) then, $P\to Q$ will be true (T). The truth table for $P\to Q$ is given below:

Note: Here, the student should always try to relate implication or conditional statements with practical examples from our day to day life for better clarity of the concept. And in any such problem where the implication sign $''\to ''$ is present then always try to write the truth table first while solving.