Implication (→)

Formal Statement

The implication of two propositions $p$ and $q$, denoted $p\to q$, is true if and only if either $p$ is false or $q$ is true. The implication $p\to q$ can be read as “If $p$, then $q$.”

Truth Table

$p$	$q$	$p\to q$
T	T	T
T	F	F
F	T	T
F	F	T

Example

If $p$ represents “I am at home” and $q$ represents “It is raining,” then $p\to q$ represents “If I am at home, then it is raining.”

Different Ways of Express Implication

Converse, Contrapositive, and Inverse

• Converse: Reverses the hypothesis and conclusion ($q\to p$).
• Inverse: Negates both the hypothesis and conclusion ($\neg p\to \neg q$).
• Contrapositive: Reverses and negates both the hypothesis and conclusion ($\neg q\to \neg p$), and is logically equivalent to the original implication.

Implication (p → q)

• The implication $p\to q$ is read as “If $p$, then $q$.”
• Example: “If it is raining, then the ground is wet.”
• Hypothesis (antecedent): $p$ (“It is raining”)
• Conclusion (consequent): $q$ (“The ground is wet”)

$p$	$q$	$p\to q$
T	T	T
T	F	F
F	T	T
F	F	T

Converse (逆命題)

• Definition: The converse of $p\to q$ is $q\to p$, where the roles of $p$ and $q$ are reversed.
• Example: “If the ground is wet, then it is raining.”
• Key Point: The converse is not logically equivalent to the original implication. The truth values of $p\to q$ and $q\to p$ may differ.
• Truth Table for Converse:

$p$	$q$	$p\to q$	$q\to p$
T	T	T	T
T	F	F	T
F	T	T	F
F	F	T	T

Inverse (否命題)

• Definition: The inverse of $p\to q$ is $\neg p\to \neg q$, which negates both the hypothesis and the conclusion of the original implication.
• Example: “If it is not raining, then the ground is not wet.”
• Key Point: The inverse is also not logically equivalent to the original implication.
• Truth Table for Inverse:

$p$	$q$	$p\to q$	$\neg p\to \neg q$
T	T	T	T
T	F	F	T
F	T	T	F
F	F	T	T

Contrapositive (對偶命題)

• Definition: The contrapositive of $p\to q$ is $\neg q\to \neg p$, which negates both $p$ and $q$ and flips their positions.
• Example: “If the ground is not wet, then it is not raining.”
• Key Point: The contrapositive is logically equivalent to the original implication. If $p\to q$ is true, then $\neg q\to \neg p$ is also true, and vice versa.
• Truth Table for Contrapositive:

$p$	$q$	$p\to q$	$\neg q\to \neg p$
T	T	T	T
T	F	F	F
F	T	T	T
F	F	T	T