Question #23825

course: abstract math
book: mathematical proofs: a transition to advanced mathematics 3rd edition
Question: For the sets A={1,2,...,10} and B={2,4,6,9,12,25}, consider the statements P: A ⊆ B. Q:|A-B|=6.
Determine which of the following statements are true.
(a) P∨Q
(b) P∨(~Q)
(c) P∧Q
(d) (~P)∧Q
(e) (~P)∨(~Q)

Expert's answer

1 is in A but not in B. So, the expression A ⊆ B is false

|A-B| = |{1, 3, 5, 7, 8, 10}| = 6

So, P is false and Q is true.

(a) false ∨ true = true

(b) false ∨ ~(true) = false ∨ false = false

(c) false ∧ true = false

(d) ~(false) ∧ true = true ∧ true = true

(e) ~(false) ∨ ~(true) = true ∨ false = true

|A-B| = |{1, 3, 5, 7, 8, 10}| = 6

So, P is false and Q is true.

(a) false ∨ true = true

(b) false ∨ ~(true) = false ∨ false = false

(c) false ∧ true = false

(d) ~(false) ∧ true = true ∧ true = true

(e) ~(false) ∨ ~(true) = true ∨ false = true

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