Day 04Part 14- Discrete mathematics for computer science – Example based on implication part 02

Day 04Part 14- Discrete mathematics for computer science – Example based on implication part 02

https://www.gyanodhan.com/video/7B4.%20GATE%20CSEIT/Discrete%20Mathematics%201/471.%20Day%2004Part%2014-%20Discrete%20mathematics%20for%20computer%20science%20-%20Example%20based%20on%20implication%20part%2002.mp4

Implication, often expressed as “if $p$, then $q$” (denoted $p\to q$), is a fundamental concept in discrete mathematics and computer science. Understanding its nuances is crucial for logical reasoning and problem-solving.

Truth Table for Implication:

$p$	$q$	$p\to q$
T	T	T
T	F	F
F	T	T
F	F	T

This table indicates that an implication $p\to q$ is false only when $p$ is true and $q$ is false; in all other cases, it is true.

Example 1:

• Let $p$: “It is raining.”
• Let $q$: “The ground is wet.”

The implication $p\to q$ translates to: “If it is raining, then the ground is wet.”

• If it is raining ( $p$ is true) and the ground is wet ( $q$ is true), the implication holds true.
• If it is raining ( $p$ is true) and the ground is not wet ( $q$ is false), the implication is false.
• If it is not raining ( $p$ is false), regardless of the ground’s condition ( $q$ true or false), the implication is considered true.

Example 2:

• $p$: “You study hard.”
• $q$: “You will pass the exam.”

The implication $p\to q$ means: “If you study hard, then you will pass the exam.”

• If you study hard ( $p$ is true) and pass the exam ( $q$ is true), the implication is true.
• If you study hard ( $p$ is true) and do not pass the exam ( $q$ is false), the implication is false.
• If you do not study hard ( $p$ is false), the implication is true regardless of whether you pass or not.

Understanding the Implication Truth Table:

The truth table for implication might seem counterintuitive, especially when $p$ is false. However, in logical terms, an implication $p\to q$ is only false when $p$ is true, and $q$ is false. In all other scenarios, it is true. This definition aligns with the principle that a false hypothesis cannot lead to a false conclusion in logical reasoning.

For a more in-depth understanding, you might find this video helpful: