Question

g Let Z be the set of all nonzero integers. (a) Use a counterexample to explain why the following statement is false: For each x 2 Z , there exists a y 2 Z such that xy D 1. (b) Write the statement in part (a) in symbolic form using appropriate symbols for quantifiers. (c) Write the negation of the statement in part (b) in symbolic form using appropriate symbols for quantifiers. (d) Write the negation from part (c) in English without usings the symbols for quantifiers.

Answer 1

(b) ∀x∈ℝ ∃ y∈ℝ: xy=1

(c)∀x∈ℝ ∄ y∈ℝ: xy=1

Explanation:

(a)For each x∈ℝ , there exists a y∈ℝ such that xy=1.

Given x=2∈ℝ

xy=1

2y=1

[TeX]y=\frac{1}{2}[/TeX]

But [TeX]y=\frac{1}{2}[/TeX]∉ℝ.

In fact, [TeX]y=\frac{1}{2}[/TeX]∈ℚ, the set of Rational Numbers.

Therefore, the statement is false.

(b) ∀x∈ℝ ∃ y∈ℝ: xy=1

(c)∀x∈ℝ ∄ y∈ℝ: xy=1

(d)For each x in the set of Real numbers , there does not exists a y in the set of real numbers such that such that xy=1.