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.

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

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

$p$	$q$	$p\to q$	$q\to p$ (Converse)	$\neg p\to \neg q$ (Inverse)	$\neg q\to \neg p$ (Contrapositive)
T	T	T	T	T	T
T	F	F	T	T	F
F	T	T	F	F	T
F	F	T	T	T	T

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

Certainly! Let’s delve into the implication operator and its related concepts: converse, inverse, and contrapositive, along with their truth tables and logical relationships.

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🔁 Implication Operator (→)

• Definition: A conditional statement of the form P → Q reads as “If P, then Q”. Here, P is the hypothesis (antecedent), and Q is the conclusion (consequent).
• Truth Table:
  

  P	Q	P → Q
  T	T	T
  T	F	F
  F	T	T
  F	F	T

  The implication P → Q is false only when P is true and Q is false; in all other cases, it is true.

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🔄 Converse, Inverse, and Contrapositive

Given the implication P → Q, we can form:

1. Converse: Q → P (“If Q, then P”)
2. Inverse: ¬P → ¬Q (“If not P, then not Q”)
3. Contrapositive: ¬Q → ¬P (“If not Q, then not P”)(Wikipedia, Wikipedia, Wikipedia)

• Truth Table:
  

  P	Q	P → Q	¬P	¬Q	¬P → ¬Q	Q → P	¬Q → ¬P
  T	T	T	F	F	T	T	T
  T	F	F	F	T	T	F	F
  F	T	T	T	F	F	T	T
  F	F	T	T	T	T	T	T

  From the table, we observe:

  • The implication and its contrapositive have identical truth values.
  • The converse and inverse also share identical truth values.(Calcworkshop, Reddit)

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🔗 Logical Equivalences

• Implication ↔ Contrapositive: Always logically equivalent.
• Converse ↔ Inverse: Always logically equivalent.
• Implication ↔ Converse: Not necessarily equivalent.
• Implication ↔ Inverse: Not necessarily equivalent.(Reddit, Lumen Learning)

This means that if the original implication is true, its contrapositive is also true, but the converse and inverse might not be.(Personal Math UBC)

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🧠 Example

Consider the statement: “If it is raining (P), then the ground is wet (Q).”(Wikipedia)

• Converse: “If the ground is wet (Q), then it is raining (P).”
• Inverse: “If it is not raining (¬P), then the ground is not wet (¬Q).”
• Contrapositive: “If the ground is not wet (¬Q), then it is not raining (¬P).”(Wikipedia)

In this scenario, the contrapositive holds the same truth value as the original statement, while the converse and inverse may not.

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For a more detailed explanation and visual guidance, you might find the following video resource helpful:

Truth Table for Implication, Converse, Inverse and Contrapositive

Feel free to reach out if you have further questions or need additional clarification on these concepts!

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

Implication – Conditional Statement p → q (p implies q) (if …