Question

Write down the negation of the following statements and determine the truth value of the negation:

a) ∀𝑥 ∈ 𝑹, 𝑥 2 + 1 ≥ 2𝑥

b) ∀𝑥 ∈ 𝑹, (𝑦 ≠→ (𝑦 + 1)/𝑦 < 1

c) ∃𝑧 ∈ 𝒁, (𝑧 𝑖𝑠 𝑜𝑑𝑑) ∨ (𝑧 𝑖𝑠 𝑒𝑣𝑒𝑛)

d) ∃𝑛 ∈ 𝑵, (𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛) ∧ (√𝑛 𝑖𝑠 𝑝𝑟𝑖𝑚𝑒)

Answer 1

a)$\exist x \isin R, x^2 +1<2x$. False, because least value of $x^2 -2x +1=(x-1)^2$ is 0 in case x=1.

b)$\exist y \isin R, y ≠ 0 ∧ (y+1)/y ≥ 1$. True, because for example with any $y>0$ , $1+1/y ≥1$.

c)$\forall z \isin Z,$ $(z \space is \space not \space odd) ∧ (z \space is \space not \space even)$. False, because if we choose any odd or even integer, and statement become false.

d)$\forall n \isin N, (n \space is \space not \space even) ∨ (√n \space is \space not \space prime)$. False, because for example n=4, in this case n is even and √n=√4=2 is prime.