Day 04Part 08-Implication operator with truth table concept of converse Cinverse and Contrapositive.

Day 04Part 08-Implication operator with truth table concept of converse Cinverse and Contrapositive.

 Implication Operator & Truth Table: Converse, Inverse, and Contrapositive

In propositional logic, the implication operator (→) is used to express a logical relationship between two statements. It is written as:

$$p\to q$$

which is read as “If $p$, then $q$“.

 Truth Table of Implication ($p\to q$)

The implication statement $p\to q$ is false only when $p$ is true and $q$ is false; otherwise, it is true.

Key Takeaways:

• If $p$ is false, the implication $p\to q$ is always true (because nothing is required from $q$).
• The only way $p\to q$ is false is when $p$ is true and $q$ is false.

 Converse, Inverse, and Contrapositive

 Converse of $p\to q$:

• The converse of an implication switches the hypothesis and conclusion: $$q\to p$$
• Example:
  • Original: “If it rains, then the ground is wet.” ($p\to q$)
  • Converse: “If the ground is wet, then it rains.” ($q\to p$) –  Not always true!

 Inverse of $p\to q$:

• The inverse negates both $p$ and $q$: $$\neg p\to \neg q$$
• Example:
  • Original: “If it rains, then the ground is wet.” ($p\to q$)
  • Inverse: “If it does not rain, then the ground is not wet.” ($\neg p\to \neg q$) – Not always true! (Ground could be wet from something else.)

 Contrapositive of $p\to q$:

• The contrapositive switches and negates both $p$ and $q$: $$\neg q\to \neg p$$
• Example:
  • Original: “If it rains, then the ground is wet.” ($p\to q$)
  • Contrapositive: “If the ground is not wet, then it did not rain.” ($\neg q\to \neg p$) –  Always true!

Key Rule:
 The contrapositive of a statement is always logically equivalent to the original statement.

 Truth Table for Converse, Inverse, and Contrapositive

Observations:

• The contrapositive always matches the original statement.
• The converse and inverse are logically equivalent but not necessarily true if the original implication is true.

 Practical Applications of Implication in Logic & Mathematics

Mathematical Proofs – Used in proving theorems (Direct & Contrapositive Proof).
Programming & AI – Logical condition checking in if-else statements.
Philosophy & Debates – Used to establish logical arguments.
Everyday Reasoning – “If you study, then you will pass” → “If you didn’t pass, you didn’t study.”

Would you like more examples or explanations?

Day 04Part 08-Implication operator with truth table concept of converse Cinverse and Contrapositive.

Truth Tables, Tautologies, and Logical Equivalences

THE CONDITIONAL TRUTH TABLE FOR THE …

B.Sc. Mathematics (Honours/Major) Class Note