Discrete Structures - Examination 2 - Fall 1997
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1. Circle T of the corresponding statement is True and F if it is False:
T F 1 + 2 + 3 + 4 + ... + 1,000 = 1,001,000
T F If A is a non-empty set, then A x A is an Equivalence Relation.
T F If f is a function whose image and range are the same set, then f is ONTO.
T F If g:A → B is a function and |g(A)| = |B|, then g is a ONE-TO-ONE function.
T F If A = {a,b,c}, then the relation R = {(a,a)} is the smallest Equivalence Relation on A.
T F |N| = |Q|.
T F If H is the Hamming distance function on binary strings, then H(10110011,11110000) = 4.
T F If f:A → B is a function and iB:B → B is the identity function on B, then (iB ° f) = f.
T F If two binary strings have the same density, then they are equal.
T F If R is an Equivalence Relation on a set A, and a,b ∈ A with [a] = [b], then (a,b) ∈ R.
2. Describe the Hamming distance function as the composition of two functions.
3. Write as a summation ranging from i = 5 to 22.
4. Let Σ = {0,1}, let d(.) be the density function on binary strings. and
let R = {(s,t) | s,t ∈ Σ4 and d(s) = d(t)}.
a. Prove that R is an Equivalence Relation.
b. What partition of Σ4 does the relation R induce?
5. Let f:R → R be the function f(x) = 5x + 9.
a. Prove that f is 1-1 and onto.
b. Find f-1(x).
6. Let f = {(1,9),(2,7),(3,5),(4,3),(5,1)} and let g = {(1,8),(3,6),(5,4),(7,2),(9,0)}. Find f−1°g−1.
7. Prove 1 of the following 2 statements using the indicated method:
a. Using Strong Induction, show that if n is an integer greater than 1, then n has a prime factor.
b. Using Weak Induction, show that an n-element set has 2n subsets.
a2
13
-----a3
24
-----a4
35
-----…a19
1820
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