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.