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

Ali Seron Jason

2 months ago

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Contents

1 Implication (Conditional Statement) in Discrete Mathematics
- 1.1 Definition:
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:

$\rightarrow Q$

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

$P$	$Q$	$\rightarrow Q$
T	T	T
T	F	F
F	T	T
F	F	T

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: $\rightarrow Q$

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: $\rightarrow Q$
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 $\rightarrow Q$ is true (based on the truth table).

Contrapositive, Converse, and Inverse

Form	Statement
Implication	$\rightarrow Q$ (If P, then Q)
Converse	$\rightarrow P$ (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 ( $\rightarrow Q$ ) is one of the fundamental logical connectives in discrete mathematics. It is widely used in mathematical proofs, computer science, and logic circuits.

Implication (Conditional Statement) in Discrete Mathematics

Definition:

Truth Table for Implication

Examples of Implication Statements

Example 1: Basic Conditional Statement

Example 2: Mathematical Implication

Example 3: False Implication

Contrapositive, Converse, and Inverse

Conclusion

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

DISCRETE MATHEMATICS FOR COMPUTER SCIENCE

Discrete Mathematics for Computer Science

DISCRETE MATHEMATICS