Reference no: EM13848688

1. Let A be a subset of the real numbers. Iva, Peter, Trefor, and Fedya tell us when they like it:

• Iva likes A if and only if ∃ a ∈R such that ∀x∈ A, a ≤ x.

 • Peter likes A if and only if ∃a ∈ A such that ∀x∈ A ,a ≤ x.

• Trefor likes A if and only if ∃a∈ R such that ∀x ∈A, a<x.
• Fedya likes A if and only if ∃a ∈A such that ∀x ∈A ,a<x.

Answer the following questions about Iva, Peter, Trefor, and Fedya. Prove your answers.

(a) Is there any of them who likes every subset of the real numbers?

(b) Is there any of them who does not like any subset of the real numbers?

(c)Who likes the empty set?

 (d) Are there two different people who like exactly the same subsets of the real numbers?

(e) Is the following statement true?

For every A ⊆ R, if Iva likes A then Peter likes A.

(f) Is the following statement true?

For every A ⊆ R, if Peter likes A then Iva likes A.