Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Which of the following arguments is valid?

A. If it rains, then the grass grows. The worms are not happy unless it rains. Therefore, If the worms are happy , then the grass grows.

B. If the wind howls, then the wolf howls. If the wind howls, then the birds sing. Therefore, if the birds sing, then the wolf howls.

C. If the sun shines, then it is day. If the stars shine, then the sun does not shine. Therefore, if the stars shine, it is not day

D. Both A and C.

share|improve this question
    
So what do you think? –  azimut Mar 14 '13 at 13:32
add comment

3 Answers

Only (A) is valid.

In options B and C the 3rd statement is not implied from the $1st$ and $2nd$ statements.

Solution:

(A) $R => G$, ~R => ~W$ (or) $W => R, hence $W => G$

(B) Wind => Wolf, Wind => Birds, but Birds !=> Wolf

(c) Sun => Day (or) ~Day => ~Sun, Stars => ~Sun, but Stars !=> ~Day

NOTE : '!=>' means 'does not imply'

share|improve this answer
add comment

A. $(U \wedge (\neg V \Rightarrow \neg U) \wedge (V \Rightarrow W)) \Rightarrow W$

True. If the worms are happy, then it rains, then the grass grows.

B. $((U \Rightarrow V) \wedge (U \Rightarrow W)) \Rightarrow (V \Rightarrow W)$

Wrong. If the birds sing, you don't know anything else. Especially, you don't know if the wind howls, only the converse is true.

C. $((U \Rightarrow V) \wedge (W \Rightarrow \neg U)) \Rightarrow (W \Rightarrow \neg U)$

Wrong. If the stars shine, then the sun does not shine, but you can't conclude the sun does not shine. Since $U \Rightarrow V$ does not imply $\neg U \Rightarrow \neg V$, but $\neg V \Rightarrow \neg U$.

share|improve this answer
add comment

If D) is valid then A) and C) are also valid; therefore there will be three valid statements.

If your don't select one of A),B) or D) then your answer is wrong.

There is only one true argument, and is not B).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.