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Discrete Structures - Fall 1993 Exam 2
1. Circle the correct answer:
i. 26 mod 7 = a. 3.714285714286 b. 0 c. 5 d. 3 e. 182
ii. 26 div 7 = a. 3.714285714286 b. 0 c. 5 d. 3 e. 182
2. Rewrite in expanded form.
3. Rewrite 3 - 3(5) + 3(52) - 3(53) +...- 3(511) in summation form.
4. Transform the summation using the rule i = k - 1.
5. Let A = {1,2,3,4,5}, B = {2,4,6,8}, and C = {0,3,6,9} be subsets of the Universal set
U = {0,1,2,3,4,5,6,7,8,9}. Find: a. (A ∩ B) ∪ (B ∪ C)C b. P(A ∩ B)
6. Use the Set Identities to show: For all sets A and B, (BC ∪ (BC ∩ AC))C = B.
(Hint: If X and Y are any sets, then X ∩ Y ⊆ X)
7. Circle T if the following statements are true or F if they are false.
T F 2 ∈ {1,2,3} T F 2 ⊆ {1,2,3}
T F {2} ∈ {1,2,3} T F {2} ∈ { {1},{2} }
T F 2 ∈ { {1},{2} } T F {2} ⊆ { {1},{2} }
8. Let Σ ={x,y} be an alphabet. Find Σ1 × Σ2 .
10. Do any 2 of the 3 proofs below. You get credit for the first two that you prove only
(so don’t do all three!).
a. Prove that if r and s are rational numbers with r < s, then there is a rational number x
such that r < x < s.
b. Prove the following statement by contradiction: If m2 is even, then m is even.
c. Prove, by Mathematical Induction, that
7m
m0=
4
∑
k2
k1+
------------
k1=
5
∑
ii 1–()
nn 1–()n1+()
3
-------------------------------------=
i2=
n
∑
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