Given this Translation scheme:

 

Nx: x is a number

Ox: x is odd

Ex: x is even

Lxy: x<y

Bxyz: x is between y and z

s(x)= the successor of x

p(x,y)=x´y

c(x)=x3

r=Rachel’s favorite number

 

How would you formalize the following sentences?

 

1. A number is odd unless it is even.

2. Every number is even or odd, but no number is both even and odd.

3. The cube of Rachel’s favorite number is even.

4. The product of any two numbers is less than the product of their successors.

5. The successor of Rachel’s favorite number is between Rachel’s favorite number and its cube.

6. Every number other than Rachel’s favorite number is less than its square.

7. The square of every number is less than its cube.

8. The product of Rachel’s favorite number and its successor is less than its cube.

9. The cube of every number greater than Rachel’s favorite number is odd.

10. Rachel’s favorite number is not the only number the cube of whose square is greater than Rachel’s favorite number.

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