Conditional Statement

Abstract

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• Given P: it rains, Q: the floor is wet
• Compound Statement P -> Q: If it rains, then the floor is wet

Warning

• If it doesn’t rain, then the floor can be either wet or no wet

• The floor MUST be wet if it rains
• Logically Equivalent to ~P OR Q
• true -> false is false, false -> true/false is true

P	Q	P → Q	~P OR Q
0	0	1	1
0	1	1	1
1	0	0	0
1	1	1	1

Terminologies

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Hypothesis/Antecedent

• After if

Conclusion/Consequent

• After then

Vacuously True

• True by default
• When the “if” part of an if-then statement is false, then statement as a whole is said to be true regardless of whether the conclusion is true of false

Implication Law

• Convert -> to OR
• Useful when we need to perform Proof by Contradiction (反证法) on Conditional Statement

Converse (相反)

• Compound Statement Q -> P: If the floor is wet, then it rains
• When the floor isn’t wet, it can still rain or doesn’t rain
• It must rain IF the floor is wet

Q	P	Q → P
0	0	1
1	0	0
0	1	1
1	1	1

Inverse (对立)

• Compound Statement ~P -> ~Q: if it doesn’t rain, then the floor isn’t wet
• When it does rain, the floor can be wet or cant be wet
• The floor must NOT BE WET IF it isn’t raining
• Logically Equivalent to Converse

P	Q	~P	~Q	~P → ~Q
0	0	1	1	1
0	1	1	0	0
1	0	0	1	1
1	1	0	0	1

Contrapositive (逆否命题)

• Compound Statement ~Q -> ~P: If the floor isn’t wet, then it doesn’t rain
• It must not rain if the floor isn’t wet
• Logically Equivalent to Standard Conditional Statement

Q	P	~Q	~P	~Q → ~P
0	0	1	1	1
1	0	0	1	1
0	1	1	0	0
1	1	0	0	1

Bi-conditional

• p <-> q, if AND only if, iff

Sufficient Condition

• r is a sufficient condition for s, r -> s

Necessary Condition

• r is a necessary condition for s, s -> r