Download 1. Translate the following statement into predicate logic: “The

1. Translate the following statement into predicate logic:
“The
multiplicative identity for the reals is unique; in other words, no number
other than 1 can serve as the identity for multiplication.”
2. Define the function even used in the book using the definition of prime
given in class as a guide.
3. Translate the following statement into predicate logic: “Among any pair
of successive numbers, one is even.” (You may use the function even to
express this.)
4. Translate the following statement into predicate logic: “There is a pair of
successive numbers neither of which is even.” (You may use the function
even to express this.)
5. Translate the following statement into predicate logic: “The equation
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x5 − 3x4 + x2 − 7x + 1 = 0 has a solution which is a real number.”
6. Translate the following statement into predicate logic: “The equation
x5 − 3x4 + x2 − 7x + 1 = 0 has a solution which is a natural number.”
7. Translate the following statement into predicate logic: “The equation
x5 − 3x4 + x2 − 7x + 1 = 0 has a solution which is a real number between
3 and 7.”
8. Translate the following statement into predicate logic: “Every solution
of the equation x5 − 3x4 + x2 − 7x + 1 = 0 is positive.”
9. Translate the following statement into predicate logic: “The natural
number k is a power of 2.”
10. Express “The natural number k is a power of 2.”
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using only the
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quantifiers ∀ and ∃ but not (+j| : . . .) or (·j| : . . .). (This is requires
some knowledge of how to factor natural numbers.)
11. Express “The natural number k is a power of 10.” using only the
quantifiers ∀ and ∃ but not (+j| : . . .) or (·j| : . . .). (This is harder than
the previous case! Do not spend too much time on it.)
12. A pair of primes is pair of two numbers p and p + 2 such that both are
primes. Translate the following statement into predicate logic: “There
are infinitely many pairs of primes”.
13. Translate the following statement into predicate logic: “No natural
number is the sum of the numbers less than itself.”
14. Translate the following statement into predicate logic: “There is a
number greater than 3 whose square is a prime.”
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