Discrete Structures - Examination 2 - Fall 2001
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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 = 500,500
T F If A is a non-empty set, and R is an Equivalence Relation on A, and a,b ∈ A, then (a,b) ∈ R.
T F If f:X → Y is a function and f(X) = Y, then f is an ONTO function.
T F If f:X → Y is a ONE-TO-ONE function then | X | = | f(X) |.
T F If f:X → Y is a function and X has 10 elements, then f has 10 elements.
T F If A is a set with 10 elements, then the smallest Equivalence Relation on A has 10 elements.
T F If f and g are functions with f:X → Y and g:Y → Z, then f ° g = g ° f.
T F If f:X → Y is a function and iY :Y → Y is the identity function on Y, then (iY ° f ) = f.
T F 1 + 10 + 100 + 1000 + ...+101000 = 101001 − 1.
T F If R is the Equivalence Relation on the Integers, Congurence Mod 7, then [23] = [1423].
2. Circle F for function, I for one-to-one, S for onto, and B for one-to-one correspondence as the properties apply
to the relations below. Circle the BEST choice for each relation, i.e. only 1 answer for each.
3. Write :
a. as a summation in i ranging from i = 7 to 38 b. as a summation in j ranging from j = 9 to 40.
4. Let Σ = {0,1}and let H be the Hamming distance function on binary strings. Consider the relation:
R = {(s,t) | s,t ∈ Σ4 and H(s,1010) = H(t,1010)}.
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) = y = 11x + 25. a. Show that that f is 1-1 and onto. b. Find f -1(x).
6. Let f = {(1,b),(2,a),(3,d),(4,e),(5,c)} and let g = {(a,8),(b,6),(c,4),(d,2),(e,0)}.
Show ( g ° f ) −1 = ( f −1 ° g−1 ).
7. Prove 1 of the following 2 statements:
a. There exists a 1-1 correspondence between A = {y | y = 3x + 2, for x ∈ N} and B = {y | y = 2x + 3, for x ∈ N}.
b. Using Weak Induction, show that 1 + 2 + 22 + 23 +...+ 2n = 2n+1 − 1.
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a. F I S B b. F I S B c. F I S B d. F I S B e. F I S B
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