DHARMPROPOSITIONAL LOGIC 107
Write the statement in symbolic form.
(i) If it rains but I stay home. I won’t be wet.
Given statement is equivalent to,
⇒ (If it rains but I stay home) “then” (I won’t be wet).
⇒ (If it rains “and” I stay home) “then” (I won’t be wet).
⇒ (R ∧ S) → (~ W)
so (R ∧ S) → ~ W will be its symbolic representation.
(ii) I’ll be wet if it rains.
Then the equivalent statement is
⇒ If wet then rains (is a meaningless sentence)
So the meaningful sentence is,
⇒ (If it rains) “then” (I will be wet).
⇒ R → W will be its symbolic representation.
(iii) If it rains and the picnic is not cancelled or I don’t stay home then I’ll be wet.
Given statement is equivalent to,
⇒ ((If it rains “and” the picnic is not cancelled) “or” (I don’t stay home)) “then” (I’ll be
wet)
⇒ ((R ∧ ~ P) ∨ ~ S ) → W
(iv)Whether or not the picnic is cancelled, I’m staying home if it rains.
Above statement is equivalent to.
⇒ (Picnic is cancelled “or” picnic is not cancelled), (I’m staying home if it rain).
⇒ (If it rain “and” (picnic is cancelled “or” picnic is not cancelled)) “then” (I’m staying
home).
Now it is easier to symbolize the sentence.
⇒ (R ∧ (P ∨ ~P) ) → S.
(v) Either, it doesn’t rain or I’m staying home.
⇒ ~ R ∨∨∨∨∨ S
(vi) Picnic is cancelled or not, I will not stay at home so I’ll be wet.
Above statement is equivalent to,
⇒ (Picnic is cancelled “or” picnic is not cancelled) “but” (I will not stay home)
“so” (I’ll be wet).
⇒ (If (picnic is cancelled “or” picnic is not cancelled) “and” (I will not stay home))
“then” (I’ll be wet).
⇒ ((P ∨ ~ P) ∧ ~ S ) → W
5.3. EQUIVALENCE OF FORMULA
Assume A and B are two statement formulas (symbolic logic) then formula A is equivalent to
formula B if and only if the truth values of formula A is same to the truth values of formula B
for all possible interpretations.
Equivalence of formula A and formula B is denoted as A ⇔ B (read as “A is equivalent
to B”).
Now we discuss a theorem that shows the equivalence of formulas. And, also purposely
we state the theorem here. As we see that basic connectors are conjunction (∧), disjunction (∨)