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Discrete Structures - Fall 1995 - Exam 2
1. Let S be the relation defined on Z as S = {(x,y)| x,y ∈ Z and x2 = y2}.
(a) Prove S is an equivalence relation.
(b) Describe the partition of Z induced by S.
2. Draw the directed graph of the relation R on A = {1,2,3,4,5,6,7,8} defined as
R = {(a,b) | a,b ∈ A and a ≡ b mod 5}.
3. Let f = {(1,d),(2,c),(3,a),(4,b),(5,e)} and g = {(a,4),(b,5),(c,1),(d,2),(e,3)} be
functions. Find: (a) f ° g(b) g ° f (c) (g ° f)−1
4. For the mappings drawn below, circle the BEST CHOICE from the list of properties:
N if the mapping is NOT a function;
F if the mapping is a function;
I if the mapping is an INJECTIVE (1-1) function;
S if the mapping is a SURJECTIVE (onto) function;
B if the mapping is a BIJECTIVE (1-1 and onto) function;
N F I S B N F I S B N F I S B N F I S B
1a 1a 1a 1a
2b 2b 2b 2b
3c 3c 3c 3c
4d 4d 4d 4
e
5. Which of the strings 01010101, 11110000, or 00001111 has the smallest
Hamming distance from the string 10110011?
6. Given Σ = {0,1} and the Equivalence Relation R on Σ8 defined as
R = {(s,t) | s,t ∈ Σ8 and the middle 6 bits of s = middle 6 bits of t},
what is [00000000]?
7. Rewrite: using the summation:
13
Σ ______
k = 4
8. Prove, by Mathematical Induction, that
1
12⋅
---------- 2
23⋅
---------- 4
34⋅
---------- 8
45⋅
---------- 16
56⋅
---------- …512
10 11⋅
----------------++++++
ii 1–()
nn 1–()n1+()
3
-------------------------------------=
i2=
n
∑
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