Conditional Statement

Abstract

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• Compound Statement in the form $P\to Q$
• Vacuously True when the Hypothesis $P$ is $0$ aka false
• Logical Equivalence to Contrapositive (逆否命题) and $\neg P\vee Q$

$P$	$Q$	$P\to Q$	$\neg P\vee Q$
0	0	1	1
0	1	1	1
1	0	0	0
1	1	1	1

Only if

”$p$ only if $q$” means “if not $q$ then not $p$” or $\neg q\to \neg p$.

Hypothesis

• Also known as Antecedent
• The part after if

Conclusion

• Also known as Consequent
• The part after then

Bi-conditional

• Represented with $\leftrightarrow$
• $P\leftrightarrow Q$ means $P$ is true if AND only if $Q$

Important

$P\leftrightarrow Q$ is logically equivalent to $(P\to Q)\wedge (Q\to P)$.

$P$ is a sufficient condition and necessary condition for $Q$ means $P$ if and only if $Q$, or $P\leftrightarrow Q$.

Sufficient Condition

• Given $P\to Q$
• $P$ is a a sufficient condition for $Q$
• If $P$ is true, $Q$ is definitely true

Necessary Condition

• Given $P\to Q$
• $Q$ is a necessary condition for $P$
• $Q$ must be true in order for $P$ to be claimed true

Vacuously True

• True by default
• When the Hypothesis is false, then statement as a whole is said to be true regardless of whether the Conclusion is true of false

Implication Law

$$P\to Q\equiv \neg P\vee Q$$

• Convert $\to$ to $\cup$

Mathematical Proof

Useful when we need to perform Proof by Contradiction (矛盾证明法) on Conditional Statement

3 Variants

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Converse (相反)

$$Q\to P$$

$P$	$Q$	$Q\to P$
0	0	1
0	1	0
1	0	1
1	1	1

• Logical Equivalence to Inverse (对立)

Inverse (对立)

$$\neg P\to \neg Q$$

$P$	$Q$	$\neg P$	$\neg Q$	$\neg P\to \neg Q$
0	0	1	1	1
0	1	1	0	0
1	0	0	1	1
1	1	0	0	1

• Logical Equivalence to Converse (相反)

Contrapositive (逆否命题)

$$\neg Q\to \neg P$$

$Q$	$P$	$\neg Q$	$\neg P$	$\neg Q\to \neg P$
0	0	1	1	1
1	0	0	1	1
0	1	1	0	0
1	1	0	0	1

• Logical Equivalence to Standard Conditional Statement