I'm self studying How to Prove book and have been working out the following problem in which I have to analyze it to logical form:Nobody in the calculus class is smarter than everybody in the discrete math classNow, this is how, I started solving it:¬(Somebody in the calculus class is smarter than everybody in the discrete math class) ¬(If x is in calculus class then x is smartert than everybody in the discrete maths class) C ( x ) = x is in calculus class. D ( y ) = y is in discrete class. S ( x , y ) = x is smarter than y ¬ ∃ x ( C ( x ) → ∀ y ( D ( y ) ∧ S ( x , y ) ) )But this is the solution given in the Velleman's book: ¬ ∃ x [ C ( x ) ∧ ∀ y ( D ( y ) → S ( x , y ) ) ]I cannot understand how that answer is correct. Can someone explain the thing I'm missing there ?

Question

I&#039;m self studying How to Prove book and have been working out the following problem in which I have to analyze it to logical form:Nobody in the calculus class is smarter than everybody in the discrete math classNow, this is how, I started solving it:¬(Somebody in the calculus class is smarter than everybody in the discrete math class) ¬(If x is in calculus class then x is smartert than everybody in the discrete maths class)  C  (  x  )  =  x is in calculus class.   D  (  y  )  =  y is in discrete class.   S  (  x  ,  y  )  =  x is smarter than y      ¬    ∃  x  (  C  (  x  )  →  ∀  y  (  D  (  y  )  ∧  S  (  x  ,  y  )  )  )But this is the solution given in the Velleman&#039;s book:      ¬    ∃  x  [  C  (  x  )  ∧  ∀  y  (  D  (  y  )  →  S  (  x  ,  y  )  )  ]I cannot understand how that answer is correct. Can someone explain the thing I&#039;m missing there ?

Oliver Shepherd · Accepted Answer

Step 1Your answer asserts that there does not exist anyone x, who, iF x is in Calculus, then (all students y are both in Discrete math and x is smarter than them.) This is clearly not what is conveyed in the original statement.Step 2What we need, essentially, is &quot;There does not exist someone x who is enrolled in Calculus AND such that, for all students y, if y is enrolled in Discrete math, then x is smarter than y.  ¬  ∃  x            (        C  (  x  )  ∧  ∀  y  (  D  (  x  )  →  S  (  x  ,  y  )  )            )

uplakanimkk · Accepted Answer

Step 1Let C denote the set of members of calculus class and let D the set of members of discrete math class.The following statements are equivalent (explore step by step) and the last one is the Velleman answer:Step 21) Nobody in the calculus class is smarter than everybody in the discrete math class2) For every person x in C there is a person y in D such that   ¬  S      (    x    ,    y    )  3)   ∀  x  ∈  C  ∃  y  ∈  D      [    ¬    S          (      x      ,      y      )        ]  4)   ∀  x      [    x    ∈    C    ⇒    ∃    y          [      y      ∈      D      ∧      ¬      S              (        x        ,        y        )            ]        ]  5)   ¬  ∃  x  ¬      [    x    ∈    C    ⇒    ∃    y          [      y      ∈      D      ∧      ¬      S              (        x        ,        y        )            ]        ]  6)   ¬  ∃  x  ¬      [    x    ∉    C    ∨    ∃    y          [      y      ∈      D      ∧      ¬      S              (        x        ,        y        )            ]        ]  7)   ¬  ∃  x      [    x    ∈    C    ∧    ¬    ∃    y          [      y      ∈      D      ∧      ¬      S              (        x        ,        y        )            ]        ]  8)   ¬  ∃  x      [    x    ∈    C    ∧    ∀    y          [      y      ∉      D      ∨      S              (        x        ,        y        )            ]        ]  9)   ¬  ∃  x      [    x    ∈    C    ∧    ∀    y          [      y      ∈      D      ⇒      S              (        x        ,        y        )            ]        ]  There is quite some redundancy here, but I hope this give you understanding about the correctness of the answer.