Problem 0: Complete the Canvas quiz "PSet 3 - Canvas".

Problem (30 points): Write each of the following in predicate logic, using only quantifiers (over the set of integers ${\bf Z}$), logical operators, arithmetic operators ($+$, $\cdot$, $-$), and relational operators ($\lt$, $\gt$, $=$). Write the negation using the same restrictions, and the additional restriction that there are no negations in front of the quantifiers. Determine which of the negation and the original statement is true (you need not give a proof).

• There is a largest even integer.
• Any integer that is a multiple of both 4 and 6 is also a multiple of 24.
• Every integer can be written as the sum of an even integer and an odd integer.
• Between any two odd integers there is some even integer.
• Between any two distinct odd integers there is some even integer.
• If two integers sum to something greater than 30, then at least one of them is greater than 15.

Problem (25 points): Write proofs of each of the following, giving a justification of each step that includes the rule of inference, logical equivalence, axiom, or theorem used (permitted theorems include rules of arithmetic or algebra, previous results, and the following: every integer is either even or odd but not both).

• For every odd integer, the next integer is even.
• If three positive integers sum to 100, then at least one of them is greater than 33.
• If the product of two integers is odd, then both integers are odd.
• The product of two consecutive integers is even.
• For any integer $x$, if $6 | x^2$, then $4 | x^3 - x^2$.