Students are asked to prove the following statement: **Theorem:** If \( A \cap B = A \cup B \), then \( \mathcal{P}(A \cup B) = \mathcal{P}(A) \). One student provides the following proof: __________________________________________________________ **Proof:** Let \( A, B \) be sets such that \( A \cap B = A \cup B \). Let \( x \subseteq \mathcal{P}(A \cup B) \). So \( x \subseteq A \cup B \) by the definition of power sets. So \( x \subseteq A \cap B \) by our assumption. So \( x \subseteq A \) and \( x \subseteq B \) by the definition of intersection. In particular \( x \subseteq A \) by specialization. Thus, \( x \subseteq \mathcal{P}(A) \) by the definition of power sets. Thus, \( \mathcal{P}(A) = \mathcal{P}(A \cup B) \). __________________________________________________________ Is the student's proof correct? If the student is correct, then fill in the needed to - make the proof completely correct - make sure each assertion made is fully justified - make the proof written in such a way that a student in the class could follow the logic and be fully convinced that the theorem is true. If the student is incorrect, then - Identify any errors in the proof above. - Explain each error. Your explanation should be written to the student who made the error, and should try to help the student understand why what they wrote is incorrect.

Consider the solution below to this: “Prove that if A is a set then so is {A} but do NOT use an argument that involves stages explicitly”. “Proof.” We know (NOTEs!) that, for any sets A and B, {A,B} is a set. But {A} ⊆ {A, B}, so {A} is a set by the subclass theorem. What EXACTLY is wrong with the proof above?

Consider the open statement P(t) which says (∃x)(tx = 20) If we extended our universal set to R, would P(π) be true?

This is in Set Theory with de natural numbers

1. Let A = {-30, -10, 10, 30, 50} and B = {-20, -10, 0, 10} be subsets of the universal set ε = {10x x € ZZ and -3≤ x <6}. (a) List the elements of An B and (AUB)'. (b) Evaluate: (i) │A\ B|; (ii) |A × B|; [1] (iii) |P(A)|; (iv) P(B) × P(A)|. [2] (c) Are the following statements are true or false? Explain your reasoning. (i) 0 СА; (ii) {0} = P(A); (iii) {10} E AUB; (iv) {10} is a proper subset of An B. (d) List the elements of P(A) P(B). [2] [2] (e) If the set C is also a subset of Ɛ, with |C| = 5, An C = {-10, 30} and BNC {-10, 20}, list the elements of C. = [2]

Prove or disprove the following statement: Let A, B be two sets such that A ∩ B = Φ.Then P(A)- P(B) ⊆P(A - B), where P(S) is the power set of a set S.

Without proving anything, determine if the following statements are true or false. For any false statements, give an counterexample. (a) A finite, nonempty set of real numbers always contains its supremum. (b) If a < L for every element a in the set A, then sup A < L. (c) If A and B are sets with the property that a < b for every a E A and b e B, then sup A < inf B. (d) If sup A = s and sup B = t, then sup(A+ B) = s+t. Here and elsewhere, the set A+B is defined as A+B = {a+b:a € A, and b e B}. (e) If sup A < sup B then there is an element of B that is an upper bound for A.