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Unformatted text preview: Exam 2 Review Questions SETS AND COUNTING METHODS QUESTIONS (1) Mark each of the following statements about sets as “True” or “False.” (a) The order of the elements does not matter. Sol: TRUE. If you changed the order of elements in a set, you would still have the same set. (b) A set is allowed to have repeated elements. Sol: FALSE. Repeated elements are not counted in a set. (c) A set cannot be empty. Sol: FALSE. There is the empty set, denoted as { } , which has no elements. (d) All subsets of a set are smaller than the set itself. Sol: FALSE. A set B is a subset of A if all the elements in B are contained in A. Thus, if A = B, then B is a subset of A (i.e. A is a subset of itself). (e) If two sets A and B are equivalent, they have to be equal. Sol: FALSE. A is equivalent to B if A has the same number of elements as B. The elements in A do not need to be the same as B for them to be equivalent. (2) What are DeMorgan’s laws? Draw Venn Diagrams to justify each of DeMorgan’s Laws. Sol: DeMorgan’s Laws say: (A ∩ B) = A ∪ B and (A ∪ B) = A ∩ B . Draw a Venn diagram with two overlapping circles inside a universal set and label the circles A and B and then number the disjoint regions. Make sure that the lefthand side of the equality corresponds to the same regions as the righthand side. (3) Among a certain group of students, there are 22 students taking math and 26 students taking physics. What are the maximum and minimum possible number of students in this group? Sol: Minimum = 26 . That would be if all 22 students who are taking math are also taking physics (so they are included in the set of 26 students taking physics). Maximum = 48 . If the sets of students taking math was completely disjoint from the set of students taking physics, then you would have 22+26 = 48 students total (the maximum). (4) License plate tags in a particular state are composed of 3 letters followed by 3 digits with repeated letters and digits allowed. How many different license plate tags can there be in this state? Sol: 26 × 26 × 26 × 10 × 10 × 10 = 26 3 × 10 3 . We use the multiplication principle because repeats are allowed and because we are picking from two different sets (letters and numbers). (5) There are 4 TAs for Math 121, “WENFANG,” “RAYBAI,” “EMEKA,” and “STEPHANIE.” Cal culate the number of distinguishable permutations of their names? Sol: “WENFANG” has 7! 2! distinguishable permutations. “RAYBAI” has 6! 2! distinguishable permutations. “EMEKA”’ has 5! 2! distinguishable permutations. “STEPHANIE” has 9! 2! distinguishable permutations. (6) What are the conditions for using the permutation formula? Sol: Pick r elements from one set of n elements where order matters and repeats are not allowed = P( n,r ) = n !...
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This note was uploaded on 12/07/2010 for the course MATH Math 121 taught by Professor Beaulieu during the Fall '10 term at UMass (Amherst).
 Fall '10
 Beaulieu
 Math, Sets, Counting, Probability

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