True/false Quiz

Algebra Level 3

How many of the following statements are true?

(i) Tom may only go to the party if he behaves. Since he was not allowed to go to the party, he must have misbehaved.

(ii) \( \forall R > 0 \ ∀N ∈ \mathbb N \ ∃n ≥ N \ such \ that \ n > R\)

(iii) Fix S ⊂ R and set T = S ∩  {1-\(\frac{1}{n}\), n = 1, 2, 3, . . . } If \(T \neq ∅\) then T has a minimum.

(iv) If S, T ⊂ R are nonempty bounded sets of real numbers then sup(S ∪ T) = max (sup S,sup T).

(v) If S, T ⊂ R are nonempty bounded sets of real numbers then sup(S ∩ T) = min(sup S,sup T).

(vi) Given any rational number q, there is a set S of irrational numbers such that

sup S = q.

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