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Math 121 Review Solutions

Math 121 Review Solutions - Exam 2 Review Questions SETS...

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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) 0 = A 0 B 0 and (A B) 0 = A 0 B 0 . 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 left-hand side of the equality corresponds to the same regions as the right-hand 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.
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