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Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.

Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.

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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”.

Truth Table for Implication

PP QQ P→QP \rightarrow Q
T T T
T F F
F T T
F F T

Key Observations:

Examples of Implication Statements

Example 1: Basic Conditional Statement

Statement: If it rains, then the ground will be wet.

Example 2: Mathematical Implication

Statement: If a number is divisible by 4, then it is even.

Example 3: False Implication

Statement: If 2 is greater than 3, then 5 is even.

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.

Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.

DISCRETE MATHEMATICS FOR COMPUTER SCIENCE

Discrete Mathematics for Computer Science

DISCRETE MATHEMATICS

Here’s a clear and simple explanation of Implication (Conditional Statements) with examples from Day 04 Part 09 – Discrete Mathematics for Computer Science.


Implication in Discrete Mathematics (Conditional Statement)

What is an Implication?

An implication is a logical statement that has the form:

“If p, then q”
Symbolically: p → q


Truth Table of Implication (p → q)

p (Hypothesis) q (Conclusion) p → q (Implication)
T T T
T F F
F T T
F F T

Only false when the first part (p) is true and second part (q) is false.


How to Read an Implication


Examples of Implication

Example 1:

If a number is divisible by 4, then it is even.

This is a true implication because every number divisible by 4 is even.


Example 2:

If x > 5, then x² > 25

Hence, the statement is logically true.


Example 3:

If 2 is odd, then 3 is even.

Even though both are wrong, the implication is logically true in propositional logic.


Contrapositive and Converse (Advanced):

Form Statement
Implication (p → q) If p, then q
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

Only contrapositive is logically equivalent to the original implication.


Summary:


Let me know if you want practice problems, truth table exercises, or a PDF notes sheet for this topic!

Day 04Part 09-discrete mathematics for computer science-Example implication conditional statement.