Another important thing to learn is, what these connectives usually translate in English to, Conditional can translate to
If P then Q
Q, if P
Q, when P
Q unless P
P only if Q
P is a sufficient condition for Q
Q is a necessary condition for P
and Biconditional can translate to
P if and only if Q
P iff Q
P is the necessary and sufficient condition for Q
Some important equivalences...
= = (this last one is known as contrapositive law)
=
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Ex-1(a)
where
S = Unpleasant smell
E = Explosive
H = Hydrogen
Ex-1(b)
Ex-1(c)
=
=
=
=
Ex-1(d) or
A means
O means x is odd
P means x is prime
Ex-2(a)
S means, Marry will sell her house
P means, she can get a good price
A means, she finds a nice apartment
Ex-2(b)
C means, having a good credit history
D means, an adequate down payment
M means, getting the mortgate
Ex-2(c) Given sentence is equivalent to, If someone does not stop John then he'll kill himself. Its logical form is..
where S means, someone stops john and
K means, john kills himself
Ex-2(d)
where D(x,y) means, x is divisible by y and
P(x) means x is prime
Ex-3
(a)
R means, its raining
W means, its windy
S means, sun is shining
(b) , clearly its the converse of (a)
(c) , its same as (a)
(d) , clearly its the converse of (a)
(e)
=
=
= , same as (a)
(f)
=
=
=
=
=
=
= , same as (a)
(g)
=
=
=
= , converse of (a)
Ex-4(a)
S means, Sales will go up
E means, Expenses will go up
H means, Boss will be Happy
Logical form of given statement is and
| S | E | H | |||||
| F | F | F | F | T | T | F | T |
| T | F | F | T | F | T | F | T |
| F | T | F | T | T | T | T | T |
| T | T | F | T | F | T | F | F |
| F | F | T | F | T | T | F | T |
| T | F | T | T | T | T | T | T |
| F | T | T | T | T | F | F | T |
| T | T | T | T | T | F | F | F |
Clearly all the places where is T, is also T . So, can be inferred from and the argument is valid.
Ex-4(b)
X means, Tax rate goes up
U means, Unemployment rate goes up
R means, there will be a recession
G means, GNP goes up
logical form of the given statement is ,
| X | U | R | G | |||||
| F | F | F | F | T | T | F | F | T |
| T | F | F | F | T | T | F | F | T |
| F | T | F | F | T | T | F | F | F |
| T | T | F | F | F | T | F | F | F |
| F | F | T | F | T | T | F | F | T |
| T | F | T | F | T | T | F | F | T |
| F | T | T | F | T | T | F | F | F |
| T | T | T | F | T | T | F | F | F |
| F | F | F | T | T | T | F | F | T |
| T | F | F | T | T | T | T | T | T |
| F | T | F | T | T | T | F | F | F |
| T | T | F | T | F | T | T | F | F |
| F | F | T | T | T | F | F | F | T |
| T | F | T | T | T | F | T | F | T |
| F | T | T | T | T | F | F | F | F |
| T | T | T | T | T | F | T | F | F |
using the same argument as earlier, its a valid argument.
Ex-4(c)
W means, Warning light will come on
P means, Pressure is too high
R means, Relief valve is clogged
logical form is ,
| W | P | R | |||
| F | F | F | F | T | T |
| T | F | F | F | F | F |
| F | T | F | F | T | F |
| T | T | F | F | F | T |
| F | F | T | F | T | T |
| T | F | T | F | F | F |
| F | T | T | T | F | F |
| T | T | T | T | T | T |
we can see an inconsistency at line 3, so its not a valid argument.
Ex-5(a)
=
=
=
=
=
=
Ex-5(b)
=
=
=
=
Ex-6(a)
=
=
=
=
=
=
=
Ex-6(b) =
Here is the proof...
=
=
=
=
Ex-7(a)
| P | Q | R | ||||||||||
| F | F | F | T | T | T | T | T | T | T | T | T | T |
| T | F | F | F | T | T | T | F | F | T | T | F | F |
| F | T | F | T | F | F | T | F | F | F | F | T | F |
| T | T | F | T | T | F | T | F | T | F | T | F | F |
| F | F | T | T | T | T | F | T | T | F | T | T | T |
| T | F | T | F | T | T | F | F | F | F | F | T | F |
| F | T | T | T | F | T | T | T | F | T | T | T | T |
| T | T | T | T | T | T | T | T | T | T | T | T | T |
Match column 8 and 13, clearly both are same.
Ex-7(b)
=
=
=
= TRUE
Ex-8
=
=
Ex-9
= , now using Ex-8
=
Ex-10
(a)
=
=
(b)
=
=
= , clearly its same as (a)
(c)
=
=
=
=
=
=
we've reduced original form to disjunctive normal form(disjunction of conjunctions) here
(d)
=
=
clearly its same as (a) and (b)
(e)
=
clearly its same as (c)
for question 4A how did you come up with that logical statement?
ReplyDelete4(a)
ReplyDeleteS means, Sales will go up
E means, Expenses will go up
H means, Boss will be Happy
From the question, take each sentence one by one..
Either sales or expenses will go up:
If sales go up then boss will be happy:
If expenses go up then boss will be unhappy:
Since question means to say all of the above, so we take the conjunction of all of the above logical statements to get
therefore sales and expenses will not both go up, meaning(yes, there is scope of ambiguity here I guess) Both of them together will not go up..so.. not of (both sales and expense going up together):
Now, to be honest, I can't really remember how I came up with it but maybe I did reverse engineer a little bit.
Here's an alternative for 7 a)
ReplyDelete(P→Q)∧(Q→R)=
(P→Q)∧(Q→R)∧[(R∨¬R)∨(¬P∨P)]
(P→Q)∧(Q→R)∧[(¬P∨R)∨(P∨¬R)]
(P→Q)∧(Q→R)∧[(¬P∨R)∨(P∨(¬Q∧Q)∨¬R)]
(P→Q)∧(Q→R)∧[(¬P∨R)∨(¬Q∨P)∧(¬R∨Q)]
(P→Q)∧(Q→R)∧(P→R)∨(Q→P)∧(R→Q)
(P→R)∧(P→Q)∧(Q→P)∨(Q→R)∧(R→Q)
(P→R)∧[P↔Q)∨(R↔Q)]
Hi Himanshu! Thanks for these incredibly helpful posts.
ReplyDeleteOn Exercise 1.d. shouldn't the correct logical form be A ∧ P → O. Disregarding A for instance, O is by no means sufficient condition for P.