Discrete Structures - Examination 2 - Spring 1996
1. Circle T of the corresponding statement is True and F if it is False:
T F If P(k) is a statement for all k ∈ N, the Natural Numbers, with P(0) being true and that P(i) being true
implies P(i+1) is true, then by the Principle of Mathematical Induction, P(k) is true for all k ∈ N.
T F If R is a reflexive relation on {1,2,3,4,5}, then (5,5) ∈ R.
T F If F = {(a,3),(b,3),(x,3),(y,3),(z,3)}, then F is a function from {a,b,x,y,z} to {1,2,3,4,5}.
T F If G:A → B is a function with G = {(1,-1),(2,-2),(3,-3),(4,-4)}, then {-1,-2,-3,-4} ⊆ B.
T F If A is a set, then the relation R = A × A is an Equivalence Relation.
T F The Integers and the Rational Numbers have the same number of elements.
T F If A and B are sets with |A| = |B|, then there exists a bijective function mapping A to B.
T F If f:A → B and g:B → C are functions, then (g ° f):A → C is a function.
T F 1 + 2 + 3 + ... + 10,000 = 5,000,500
T F Let A be a set and R be an Equivalence Relation on A that induces the partition of A into non-empty
subsets B and C. If b ∈ B, then the equivalence class, [b] = B.
2. Which of the bytes 10101010, 10011001, 00001111, and 11110000 has least Hamming Distance from 11011011?
3. Write 1 − 5 + 52 − 53 + ... − 513 in summation notation ranging from i = 8 to i = 21.
4. If R is the Equivalence Relation given by R = {(a,b) | a,b ∈ {1,2,...,100} and a ≡ b mod 17}, what is [3]?
5. Let R be a relation on {1,2,3,4,5} given by R = {(1,1),(1,3),(2,2),(2,3),(3,2),(4,5)}. What elements is R missing to
be an Equivalence Relation?
6. Let f:{1,2,3,4,5} to {1,3,5,7,9} be the function f = {(1,3),(2,1),(3,9),(4,7),(5,5)} and
let g:{1,3,5,7,9} to {0,2,4,6,8} be the function f = {(1,8),(3,6),(5,4),(7,2),(9,0)}. Find (g°f)−1.
7. Let Σ = {0,1}. Graph a bijective function to show that Σ3 and Σ × Σ × Σ have the same cardinality.
8. Prove 3 of the following 4 statements using the indicated method:
a. Using Strong Induction, show that every Integer greater than 1 can be factored as the product of primes.
b. Using Mathematical Induction, show that a set with n elements has 2n subsets.
c. Congruence mod 5 is an Equivalence Relation on the Integers.
d. Prove the function f:R → R given by f(x) = 7x + 2 is a one-to-one and onto function.
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