Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.
Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.
Contents [hide]
- 1 Implication (Conditional Statement) in Discrete Mathematics
- 2 Truth Table for Implication
- 3 Examples of Implication Statements
- 4 Contrapositive, Converse, and Inverse
- 5 Conclusion
- 6 Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.
- 7 DISCRETE MATHEMATICS FOR COMPUTER SCIENCE
- 8 Discrete Mathematics for Computer Science
- 9 DISCRETE MATHEMATICS
Implication (Conditional Statement) in Discrete Mathematics
Definition:
An implication (also called a conditional statement) is a logical statement of the form:
P→QP \rightarrow Q
which is read as “If P, then Q” or “P implies Q”.
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PP is called the antecedent (hypothesis or premise).
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QQ is called the consequent (conclusion).
Truth Table for Implication
| PP | P→QP \rightarrow Q | |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Key Observations:
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An implication is false only when P is true and Q is false.
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If PP is false, the statement is always true, regardless of QQ.
Examples of Implication Statements
Example 1: Basic Conditional Statement
Statement: If it rains, then the ground will be wet.
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PP: It rains.
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QQ: The ground is wet.
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Logical form: P→QP \rightarrow Q
Example 2: Mathematical Implication
Statement: If a number is divisible by 4, then it is even.
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PP: A number is divisible by 4.
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QQ: The number is even.
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Logical form: P→QP \rightarrow Q
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True, since every number divisible by 4 is always even.
Example 3: False Implication
Statement: If 2 is greater than 3, then 5 is even.
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PP: 2 > 3 (False)
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QQ: 5 is even (False)
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Since PP is false, the statement P→QP \rightarrow Q is true (based on the truth table).
Contrapositive, Converse, and Inverse
| Form | Statement |
|---|---|
| Implication | P→QP \rightarrow Q (If P, then Q) |
| Converse | Q→PQ \rightarrow P (If Q, then P) |
| Inverse | ¬P→¬Q\neg P \rightarrow \neg Q (If not P, then not Q) |
| Contrapositive | ¬Q→¬P\neg Q \rightarrow \neg P (If not Q, then not P) |
Contrapositive is always logically equivalent to the original implication.
Conclusion
Implication (P→QP \rightarrow Q) is one of the fundamental logical connectives in discrete mathematics. It is widely used in mathematical proofs, computer science, and logic circuits.