Rain, Rain, Go Away

Logic Level 2

$\text{If students go to the lab, then it is raining.}$

Which statement below is logically equivalent to the statement above?

A) If students do not go to the lab, then it is not raining.
B) If students do not go to the lab, then it is raining.
C) If it is raining, then students go to the lab.
D) If it is not raining, then students do not go to the lab.

B

B , C , D

A , B , C

A , D

None of these

A

D

C

2 solutions

Kay Xspre
Jan 17, 2016

Let the statement "Student go to the lab" be P, and "it is raining" be Q. The notation is $P\Rightarrow Q$ . Now, put other choices into the notation

A) $\sim P\Rightarrow\:\sim Q$

B) $\sim P\Rightarrow Q$

C) $Q\Rightarrow P$

D) $\sim Q\Rightarrow\:\sim P$

The very basic approach to find the equivalence to the original statement is to check every truth value possible for smaller statement, but in this short solution, we will set P as True and Q as False (hence $P\Rightarrow Q$ is false )

A) $\sim P\Rightarrow\:\sim Q$

B) $\sim P\Rightarrow Q$

For the first two, since P is true, then ~P is false, but since $F\Rightarrow (T/F)$ will always be true, it contradicts our position that the sentence is false, and hence these two are not the correct choice

C) $Q\Rightarrow P$

Similarly to the above two, but we only switch Q (which is false) to the cause. This one is also not correct.

D) $\sim Q\Rightarrow\:\sim P$

This is the only correct option, which can be proven with truth table. Its properties is known as contraposition .

One can even prove that $(P \Rightarrow Q) \equiv (\sim Q \Rightarrow \sim P)$ using a truth table. This is a famous equivalence between an implication and its contrapositive.

Venkata Karthik Bandaru - 5 years, 4 months ago

Cause and effect. No effect means no cause.

Lu Chee Ket - 5 years, 4 months ago

What I realize are just:

Not (P AND Q) = Not P OR Not Q

Not (P OR Q) = Not P AND Not Q

https://en.wikipedia.org/wiki/Material_conditional

The website above may suggest that A, B, C and D are all True. "None of these" may be its implication, as "If ~ then" is considered equivalent to "Implication". I am just doubted by this question and I cannot be certain about its correct answer.

Lu Chee Ket - 5 years, 4 months ago

This question is mainly not asking us to find the implication, but rather to ask that which statement is logically equivalent (that is to say, have the same truth value for all possible cases). Upon checking, not every inverse case gives the same (for example, if P is true and Q is false, then $P\Rightarrow Q$ is false but $Q\Rightarrow P$ is true; this means $P\Rightarrow Q$ and $Q\Rightarrow P$ is not logically equivalent)

Kay Xspre - 5 years, 4 months ago

All right! I had just learnt these. Thanks!

Lu Chee Ket - 5 years, 4 months ago

All I noticed is p $\implies$ -q and q $\implies$ -p are taken as logically equivalent. p $\implies$ q and -q $\implies$ -p should also be the same.

Lu Chee Ket - 5 years, 4 months ago

This is a hypothetical qn. where there is causality between rain and students going to lab. The definition of 'then' is equivalent to afterwards and therefore. This show that raining comes after going to lab. B,C,D are wrong and A is the correct answer that fits the logic.

Zhi Wei - 5 years, 4 months ago

Rushil Gholkar
Jan 31, 2016

Essentially went through thinking about each option... worth noting that (D) (correct) is known as the contrapositive. Where A implies B, Not B implies Not A.

I am uncomfortable with the initial proposition: student actions are NOT in fact causative of rainfall - so the logical conclusions derived from the proposition are themselves lacking standing.

Brian Whatcott - 5 years, 4 months ago

Rain, Rain, Go Away

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