Analyze the logical form of the following statement: a) If it is raining, then i
ID: 3674567 • Letter: A
Question
Analyze the logical form of the following statement:
a) If it is raining, then it is windy and the sun is not shining.
Now analyze the following statements. Also, for each statement determine whether the statement is equivalent to either statement (a) or its converse.
b) It is windy and not sunny only if it is raining.
c) Rain is a sufficient condition for wind with no sunshine.
d) Rain is a necessary condition for wind with no sunshine.
e)It's not raining, if either the sun is shining or it's not windy.
f) Wind is a necessary condition for it to be rainy and so is a lack of sunshine.
g) Either it is windy only if it is raining or it is not sunny only if it is raining.
Explanation / Answer
R = It is raining. W = It is windy. S = Sun is shining.
(a)if it is raining,then it is windy and the sun is not shining.
R=>W^~S
(b)It is windy and not sunny only if it is raining.
W^~S=>R
This is converse to (a)
(c)Rain is a sufficient condition for wind with no sunshine.
Rain is sufficent condition for Wind means R=>S
R=>W^~S
This is Equivalent to (a)
(d)Rain is a necessary condition for wind with no sunshine
Rain is necessary condition for wind means W=>R
W^~S=>R
This is A converse of (a)
(e)Its Not Raining,if either the sun is shining or its not windy.
~(S V ~W) V ~R
(~S^W)V ~R
~RV(~S^W)
R=>(~S^W)
This is Equivalent to (a)
(f)Wind is a necessary condition for it to be rainy,and so is a lack of sunshine.
(R=>W)^(R=>~S)
(~RVW)^(~RV~S)
((~RVW)^(~R))V((~RVW)^(~S))
((~R)V(W^~R))V((~RVW)^(~S))
((~R)V(~R^W))V((~RVW)^(~S))
~RV((~RVW)^~S)
~RV((~R^~S)V(~S^W))
~RV(~S^W)
R=>~S^W
it is Equivalent to (a)
(g)Either it is windy only if it is raining,or it it not sunnt only if it is raining
(W=>R) V (~S=>R)
(~WVR) V (SVR)
(~WVRVSV R)
R~W V S
~R =>~WVS
it is converse of (a)
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.