Question

write the following sentences as quantified logical statements, using the universal and existential quantifiers, and defining predicates as needed. Second, write the negations of each of these statements in the same way. Finally, choose one of these statements to prove. If it is true, prove it, and if it is false, prove its negation. Your proof need not use symbols, but can be a simple explanation in plain English. 1. If m and n are positive integers and mn is a perfect square, then m and n are perfect squares. 2. The difference of the squares of any two consecutive integers is odd. 3. For all nonnegative real numbers a and b, sqrt(ab) = sqrt(a) * sqrt(b). (Note that if x is a nonnegative real number, then there is a unique nonnegative real number y, denoted sqrt(x), such that y^2 = x.)

Answer #1

Write the contrapositive statements to each of the following.
Then prove each of them by proving their
respective contrapositives. In both statements assume x
and y are integers.
a. If the product xy is even, then at least one
of the two must be even.
b. If the product xy is odd, then both x and y
must be odd.
3. Write the converse the following statement.
Then prove or disprove that converse depending on whether it is
true or not. Assume x...

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

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