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:

$$P\to Q$$

which is read as “If P, then Q” or “P implies Q”.

• $P$ is called the antecedent (hypothesis or premise).

• $Q$ is called the consequent (conclusion).

Truth Table for Implication

Key Observations:

• An implication is false only when P is true and Q is false.

• If $P$ is false, the statement is always true, regardless of $Q$.

Examples of Implication Statements

Example 1: Basic Conditional Statement

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

• $P$: It rains.

• $Q$: The ground is wet.

• Logical form: $P\to Q$

Example 2: Mathematical Implication

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

• $P$: A number is divisible by 4.

• $Q$: The number is even.

• Logical form: $P\to Q$

• 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.

• $P$: 2 > 3 (False)

• $Q$: 5 is even (False)

• Since $P$ is false, the statement $P\to Q$ is true (based on the truth table).

Contrapositive, Converse, and Inverse

Contrapositive is always logically equivalent to the original implication.

Conclusion

Implication ($P\to Q$) 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.

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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

• p is called the antecedent (or hypothesis).

• q is called the consequent (or conclusion).

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Truth Table of Implication (p → q)

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

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How to Read an Implication

• p → q means “If p happens, then q will happen.”

• Example:

  • “If it rains, then the ground will be wet.”
    → Rain → Wet Ground

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Examples of Implication

Example 1:

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

• p: A number is divisible by 4.

• q: The number is even.

• Implication: p → q

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

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Example 2:

If x > 5, then x² > 25

• Let’s test:

  • x = 6 → x² = 36 → True

  • x = 4 → x > 5 is false → So implication is true (doesn’t violate)

Hence, the statement is logically true.

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Example 3:

If 2 is odd, then 3 is even.

• p: 2 is odd → False

• q: 3 is even → False

• So p → q = True (because p is false)

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

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Contrapositive and Converse (Advanced):

Only contrapositive is logically equivalent to the original implication.

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Summary:

• Implication expresses a condition: “If this, then that.”

• False only when the first part is true, and the second is false.

• Used widely in mathematical proofs, algorithms, and logic gates.

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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.