Introduction to Math Reasoning, Spring 08
Grinshpan
QUIZ 3 ANSWERS
A) “Any natural number squared is a sum of squares of two natural
numbers.” (This is not so.)
B) “There exist a positive and an irrational x such that for any
integer n the fractional part of nx is greater than or equal to .”
COMMENT. The original statement is true. Its negation is false.
C) Recall that “X ⇒ Y ” and “¬ X OR Y ” are equivalent.
Hence so are “X ⇒ (Y ⇒ Z)” and “¬ X OR (¬ Y OR Z)”.
Parentheses in the latter statement can be dropped.
Applying ¬ , we obtain “X AND Y AND ¬ Z”.
COMMENT. X ⇒ ¬ Y is not the negation of X ⇒ Y .
Indeed, both statements are true whenever X is false.
Verify that X AND ¬ Y is the right negation.
HOMEWORK 3
a. Wrong. Not any pair (x0 , y 0 ) must be in T .
b. Correct. This would imply that S ⊆ T .
c. Wrong. This would imply that T ⊆ S, the opposite inclusion.
d. Wrong. This is only good for (1, 2).
e. Correct. Note the notational difference with b.