Math 220 Assignment 1
Due September 18th
Definition 1. Let n, d be integers. Then there exists integers q and r such
that n = qd + r, and 0 ≤ r < d. We call q the quotient, r the remainder. When
r = 0, we say n is d-divisible. For example, being 2-divisible is the same thing
as being even. Finally, n%d is defined to be r.
1. For each of the following statements, write it formally (to include universal and existential quantifiers, and other logical operators). For the last
problem, write out the statement, replacing ∀ and ∃ with English words.
(a) 1 is the smallest positive integer.
(b) There is a smallest positive rational number.
(c) If a function is continuous at x = c, then that function is differentiable
at x = c.
(d) Every nonnegative number is equal to a sum of four squares (for
example, 31 = 52 + 22 + 12 + 12 ).
(e) A function is uniformly continuous on [a, b] if ∀e > 0, ∃d > 0 3
∀x, y ∈ [a, b], |x − y| < d → |f (x) − f (y)| < e.
2. Write down the negation of each statement from Problem 1.
3. Let d be an integer. Consider the statement P =‘the sum of a pair of
3-divisible numbers is 6-divisible’.
(a) Rewrite the statement P formally, using variables, quantifiers, and
other logical operations.
(b) Write down the negation of the statement P .
(c) Prove or disprove and salvage statement P .
4. Let d be an integer.
(a) Consider the statement ‘the product of two d-divisible numbers is
d-divisible’. Rewrite the statement using quantifiers.
(b) Prove, or disprove and salvage the statement.
(c) What is the converse of the statement? Prove, or disprove the converse.
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5. Let d be an even number, n be an integer, and let r denote the remainder
for n divided by d. Prove that n is even if and only if n%d is even. Remark:
n%d is remainder for n divided by d. When d = 10, this gives the familiar
rule that an integer is even if and only if its ‘ones place’ is 0, 2, 4, 6 or 8.
Is the statement still true when d is odd?
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