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C) If x ∈ A \B, then x ∈ A or x ∉ B. (d) In proving S ⊆ T, one should avoid beginning with “Let x ∈ S,” because this assumes that S is nonempty. 3. Let A = {2, 4, 6, 8}, B = {6, 7, 8, 9}, and C = {2, 8}. Which of the following statements are true? (a) {8, 7} ⊆ B (b) {7} ⊆ B ∩ C (c) (A \B ) ∩ C = {2} (d) C \ A = ∅ (e) ∅ ∈ B (f ) A ∩ B ∩ C = 8 (g) B \ A = {2, 4} (h) (B ∪ C) \ A = {7, 9} 4. Let A = {2, 4, 6, 8}, B = {6, 8, 10}, and C = {5, 6, 7, 8}. Find the following sets. (a) A ∩ B (b) A ∪ B (c) A \B (d) B ∩ C (e) B \C (f ) (B ∪ C )\A (g) (A ∩ B )\C (h) C \(A ∪ B) 5.

B) You can use (i) to prove (b) is true. 9. (a) False; (b) True; (c) True; (d) False. 11. (a) True; (b) False; (c) True; (d) False; (e) True; (f ) True. 13. (a) ∀ x, f (−x) = f (x); (b) ∃ x f (−x) ≠ f (x). 15. (a) ∀ x and y, x ≤ y ⇒ f (x) ≤ f ( y). (b) ∃ x and y x ≤ y and f (x) > f ( y). 17. (a) ∀ x and y, f (x) = f ( y) ⇒ x = y. (b) ∃ x and y f (x) = f ( y) and x ≠ y. 19. (a) ∀ ε > 0, ∃ δ > 0 ∀ x ∈ D, | x − c | < δ ⇒ | f (x) − f (c)| < ε. (b) ∃ ε > 0 ∀ δ > 0, ∃ x ∈ D | x − c | < δ and | f (x) − f (c)| ≥ ε.

For example, the statement ∀ x, x2 = x is true in the context of the positive numbers but is false when considering all real numbers. Similarly, ∃x x2 = 25 and x < 3 is false for positive numbers and true for real numbers. When you learn about set notation, it will become easier to be precise in indicating the context of a particular quantified statement. For now, we have to write it out with words. To prove a universal statement ∀ x, p (x), we let x represent an arbitrary member from the system under consideration and then show that statement p (x) is true.