select the best translation into predicate logic, using the following translation key:
a: one
b: two
c: three
f(x): the successor of x
g(x,y): the sum of x and y
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-If the sum of the successors of any two numbers is divisible by three then the sum of two and the sum of the original two numbers is divisible by three.
A) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(x,y)c ⊃ Dg(b,x,y)c]}
B) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(x),f(y))c ⊃ Dg(b,g(f(x),f(y)))c]}
C) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(x),f(y))c ⊃ Dg(c,g(x,y))b]}
D) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(x),f(y))c ⊃ Dg(b,g(x,y))c]}
E) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(a),f(b))c ⊃ Dg(b,g(a,b))c]}

Question

select the best translation into predicate logic, using the following translation key: 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y 
Gxy: x is greater than y
-If the sum of the successors of any two numbers is divisible by three then the sum of two and the sum of the original two numbers is divisible by three.
A) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(x,y)c ⊃ Dg(b,x,y)c]}
B) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(x),f(y))c ⊃ Dg(b,g(f(x),f(y)))c]}
C) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(x),f(y))c ⊃ Dg(c,g(x,y))b]}
D) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(x),f(y))c ⊃ Dg(b,g(x,y))c]}
E) (∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(a),f(b))c ⊃ Dg(b,g(a,b))c]}

Quizplus · Accepted Answer

- The statement starts with a conditional "if". Therefore, we need to use the symbol "⊃" to translate the implication.
- The statement talks about the sum of the successors of any two numbers. We can use the function symbol "f(x)" to represent the successor of a number.
- The statement talks about the sum of two numbers. We can use the function symbol "g(x,y)" to represent the sum of two numbers.
- The statement talks about divisibility. We can use the predicate symbol "Dxy" to represent divisibility.
- The statement talks about the number "two". We can use the constant symbol "b" to represent "two".
- The statement talks about the number "three". We can use the constant symbol "c" to represent "three".
- Putting it all together, we get: 
(∀x)(∀y){(Nx • Ny) ⊃ [Dg(f(x),f(y))c ⊃ Dg(b,g(x,y))c]}

Select the Best Translation into Predicate Logic, Using the Following