Midterms Round 2

Week 8

Sunday

There will be two midterms for me next week: CSC165 and CSC148. And I was too busy to wrote any blog for the past two weeks due to the amount of assignments I had. But in this blog, I want to take chance to talk about Assignment 2 which I had last week.
Overall, the assignment was not that hard and it took me about 2.5 hours to finished it and checked for a second time. However, for question 5 in the assignment:

4. Prove or disprove: for all quadruples of positive real number w, x, y, z, if w / x < y / z, then:

( w / x < ( w + y ) / ( x + z ) ) ^ ( ( w + y ) / ( x + z ) < y / z )

A: Prove/ true

assume ∀ w, x, y, z ∈ R, w > 0, x > 0, y > 0, z > 0

and w / x < y / z

=> ( w / x < ( w + y ) / ( x + z ) ) ^ ( ( w + y ) / ( x + z ) < y / z )

assume ( w / x < ( w + y ) / ( x + z ) ) 

=> ( ( w / x ) * ( x ( x + z ) ) ) < ( ( w + y ) / ( x + z ) * ( x ( x + z ) )

then    = ( x + 2 ) w < x ( w + y )

             = x w + z w < x w + x y

             = z w < x y

             = z w / x < y

             = w / x < y / z

therefore, it is true

assume ( ( w + y ) / ( x + z ) < y / z )

=> ( ( w + y ) / ( x + z ) * z ( x + z ) ) < ( y / z * ( z ( x + z ) ) )

then    = ( w + y ) z < y ( x + z )

             = w z + z y < y x + z y

             = w z < y x

             = w / x < y /z

therefore, it is true

therefore, ∀ w, x, y, z ∈ R, w > 0, x > 0, y > 0, z > 0, w / x < y / z =>

                   ( w / x < ( w + y ) / ( x + z ) ) ^ ( ( w + y ) / ( x + z ) < y / z )

therefore, it is true

The above was my solution. I also checked the sample solution. But it is much different from mine: I separate the two parts and then solved each one of them. And it also seems that I have so much more steps than the sample solution because I thought just in case that I would lose marks for not having enough steps and explain my answer clearly.

Therefore, my question is that does my solution to the question look right? And I also have noticed that my writing style in the problem-solving is not exactly like the sample solutions nor notes, does it matter a whole lot and might causing me to lose marks in an exam?

Thank you very much for your time!

Roland Qiyang Liu