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.
Contents
- 1 Implication Operator & Truth Table: Converse, Inverse, and Contrapositive
- 2 Truth Table of Implication (p→qp \rightarrow qp→q)
- 3 Converse, Inverse, and Contrapositive
- 4 Converse of p→qp \rightarrow qp→q:
- 5 Inverse of p→qp \rightarrow qp→q:
- 6 Contrapositive of p→qp \rightarrow qp→q:
- 7 Truth Table for Converse, Inverse, and Contrapositive
- 8 Practical Applications of Implication in Logic & Mathematics
- 9 Day 04Part 08-Implication operator with truth table concept of converse Cinverse and Contrapositive.
- 10 Truth Tables, Tautologies, and Logical Equivalences
- 11 THE CONDITIONAL TRUTH TABLE FOR THE …
- 12 B.Sc. Mathematics (Honours/Major) Class Note
- 13 🔁 Implication Operator (→)
- 14 🔄 Converse, Inverse, and Contrapositive
- 15 🔗 Logical Equivalences
- 16 🧠 Example
- 17 Day 04Part 08-Implication operator with truth table concept of converse Cinverse and Contrapositive.
- 18 Implication – Conditional Statement p → q (p implies q) (if …
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→qp \rightarrow q
which is read as “If pp, then qq“.
Truth Table of Implication (p→qp \rightarrow q)
The implication statement p→qp \rightarrow q is false only when pp is true and qq is false; otherwise, it is true.
pp | p→qp \rightarrow q | |
---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Key Takeaways:
- If pp is false, the implication p→qp \rightarrow q is always true (because nothing is required from qq).
- The only way p→qp \rightarrow q is false is when pp is true and qq is false.
Converse, Inverse, and Contrapositive
Converse of p→qp \rightarrow q:
- The converse of an implication switches the hypothesis and conclusion: q→pq \rightarrow p
- Example:
- Original: “If it rains, then the ground is wet.” (p→qp \rightarrow q)
- Converse: “If the ground is wet, then it rains.” (q→pq \rightarrow p) – Not always true!
Inverse of p→qp \rightarrow q:
- The inverse negates both pp and qq: ¬p→¬q\neg p \rightarrow \neg q
- Example:
- Original: “If it rains, then the ground is wet.” (p→qp \rightarrow q)
- Inverse: “If it does not rain, then the ground is not wet.” (¬p→¬q\neg p \rightarrow \neg q) – ❌ Not always true! (Ground could be wet from something else.)
Contrapositive of p→qp \rightarrow q:
- The contrapositive switches and negates both pp and qq: ¬q→¬p\neg q \rightarrow \neg p
- Example:
- Original: “If it rains, then the ground is wet.” (p→qp \rightarrow q)
- Contrapositive: “If the ground is not wet, then it did not rain.” (¬q→¬p\neg q \rightarrow \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
pp | p→qp \rightarrow q | q→pq \rightarrow p (Converse) | ¬p→¬q\neg p \rightarrow \neg q (Inverse) | ¬q→¬p\neg q \rightarrow \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.
🔁 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.
🔄 Converse, Inverse, and Contrapositive
Given the implication P → Q, we can form:
- Converse: Q → P (“If Q, then P”)
- Inverse: ¬P → ¬Q (“If not P, then not Q”)
- 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)
🔗 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)
🧠 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.
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!