Question

 

1. Write the negation of each statement:

(a) ∀? ∈ ℝ, ∃? ∈ ℝ such that ? < ? 2.

(b) ∀? ∈ ℚ, ∃?, ? ∈ ℕ such that ? = ??.

(c) ∀ even integers ?, ∃ an integer ? such that ? = 2?.

(d) ∃? ∈ ℝ such that for all real numbers ?, ? + ? = 0.

(e) ∃?, ? ∈ ℝ such that if ? < ? then ? 2 < ? 2.

(f) ∀? > 0, ∃? > 0 such that ∀ ? ∈ (? − ?, ? + ?), |?(?) − ?| < ?. Remember this definition from Calculus?!

2. Which of the following statements are true and which are false. Justify your answers. Hint: This is a great problem. You need to be particularly careful about the order of the quantifiers (∀ and ∃).

(a) ∃? ∈ ℝ such that ∀? ∈ ℝ ? 2 + ? 2 = 9.

(b) ∀? ∈ ℝ, ∃? ∈ ℝ such that ? 2 < ? +1.

(c) ∀? ∈ ℤ +, ∃? ∈ ℤ + such that ? = ? + 1.

(d) ∀? ∈ ℤ, ∃? ∈ ℤ such that ? = ? + 1.

(e) ∃? ∈ ℝ such that ∀? ∈ ℝ, ? = ? + 1.

(f) ∀? ∈ ℝ +, ∃? ∈ ℝ + such that ?? = 1.

(g) ∀? ∈ ℝ, ∃? ∈ ℝ such that ?? = 1.

(h) ∀? ∈ ℤ + and ∀? ∈ ℤ +, ∃? ∈ ℤ + such that ? = ? − ?.

(i) ∀? ∈ ℤ and ∀? ∈ ℤ, ∃? ∈ ℤ such that ? = ? − ?.

(j) ∃? ∈ ℝ + such that ∀? ∈ ℝ +, ?? < ?.

(k) ∀? ∈ ℝ +, ∃? ∈ ℝ + such that ?? < ?.

3. Determine whether the following arguments are valid or invalid (specify if by converse or inverse error).

(a) The product of two rational numbers is rational. ?? is rational. Therefore, ? and ? are rational.

(b) If ? and ? are odd, then their sum is even. ? and ? are odd. Therefore their sum is even.

(c) If a ? × ? matrix, ?, has ? distinct eigenvalues, then it has ? linearly independent eigenvectors. ? does not have ? linearly independent eigenvalues. Therefore, ? does not have ? distinct eigenvalues.

4. Use Venn-Diagram to determine if the following arguments are valid:

(a) Everyone taking discrete mathematics can think logically. Everyone who likes ice cream can think logically. Everyone taking discrete mathematics likes ice cream.

(b) All students who took Linear Algebra took Calculus. All students who took Calculus to know how to integrate. All students who took Linear Algebra know how to integrate.

Answer 1