Answer
a. Statement: For every odd integer n, there is an integer k such that n = 2k + 1.
Given any odd integer, there is another integer for which the given integer equals twice the other integer plus 1.
Given any odd integer n, we can find another integer k so that n = 2k + 1.
An odd integer is equal to twice some other integer plus 1.
Every odd integer has the form 2k + 1 for some integer k.
b. Negation: ∃ an odd integer n such that ∀ integers k, n $\neq$ 2k + 1.
There is an odd integer that is not equal to 2k + 1 for any integer k.
Some odd integer does not have the form 2k + 1 for any integer k.
Work Step by Step
Negation of multiply-quantified statement:
~($\forall x$ in D, $\exists y$ in E such that P(x,y)) $\equiv$ $\exists x$ in D such that $\forall y$ in E, ~P(x,y)
