1) Let A and B be two sets. (a) Show that (A^c ∩ B^c)^c = A ∪ B and (A^c ∪ B^c)^c = A ∩ B. (b) Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up. Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of the equalities in part (a), and verify that the equalities hold.
1) Let A and B be two sets. (a) Show that (A^c ∩ B^c)^c = A ∪ B and (A^c ∪ B^c)^c = A ∩ B. (b) Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up. Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of the equalities in part (a), and verify that the equalities hold.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
1)
Let A and B be two sets.
(a) Show that (A^c ∩ B^c)^c = A ∪ B and (A^c ∪ B^c)^c = A ∩ B.
(b) Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up. Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of the equalities in part (a), and verify that the equalities hold.
2)Let A and B be two sets with a finite number of elements. Show that the number of elements in A ∩ B plus the number of elements in A ∪ B is equal to the number of elements in A plus the number of elements in B.
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