Download dt248 dm review fall 2015

DT6248 Discrete Maths Assessment Review 2015
1) Consider the following sets:
{w},{ y, w, z},{w, y, z},{ y, z, w},{w, x, y, z},{z, w}
Which of them are equal to A = {w, y, z} ?
2) Define, with examples, the universal set U.
3) Prove that if A is a subset of the null set  , then A =  .
4) Consider the following sets, A = {1,2,3,4}, B = {2,4,6,8}, C = {3,4,5,6} and
Universal set U = {1,2,3,…,9}.
Find  A  B  \C and (A \ B)c
5) Shade the set Ac   B \ C 
6) Suppose A  B . Show that n( A  B)  n( B) and n( A  B)  n( A) .
7) E = [{1,2,3},{2,3},{a,b}] Find the power set, (E) of E.
8) Determine whether each function is one-to-one. The domain of each function is
the set of all real numbers. If the function is not one-to-one, prove it. Determine
whether the function is onto the set of all real numbers. If the function is not
onto, prove it.
a) f ( x)  3x 2  3x  1
b) f ( x)  3x  2
9) Find the inverse of the following functions:
a)
b)
f ( x)  4 x  2, X = set of all real numbers
f ( x)  4 x3  5, X = set of all real numbers
10) Decompose f ( x) 
1
into a simpler function.
2 x2
11) Let f be the function from X = {0,1,2,3,4} to X defined by
f ( x)  4 x mod 5 Write f as a set of ordered pairs and draw the arrow
diagram of f. Is f one-to-one? Is f onto?
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12) Let R be the relation on A = {1,2,3,4} defined by “x is less than y”. Write R as a
set of ordered pairs.
13) Find the inverse of R in problem 12.
14) Let R be the relation on the set N of positive integers defined by the equation
x 2  2 y  100. Find the domain of R and the equation for R-1.
15) Draw the directed graph of the relation T on X = {1,2,3,4} defined by
T = {(1,1), (2,2), (2,3),(3,2),(4,2),(4,4)}.
16) Let A = {1,2,3,4}, B = {a,b,c,d}, and C = {x,y,z}. Consider the relations R from
A to B and S from B to C defined by
R = {(1,a),(2,d),(3,a),(3,b),(3,d)}, S = {(b,x), (b,z), (c,y), (d,z)}
Find the composition R S .
17) Determine when relation on a set A is (a) not reflexive (b) not symmetric (c) not
transitive.
18) (i) Let R, S, T be the relation on A = {1,2,3} defined by
R = {(1,1),(2,2),(3,3)}, S = {(1,2),(2,1),(3,3)}, T = {(1,2),(2,3),(1,3)}
Determine which relations are (a) reflexive. (b) symmetric (c) transitive
(ii) Determine if S : x  y  10 is (a) reflexive. (b) symmetric (c) transitive
19) Find all partitions of S = {1,2,3}.
20) Is  x : x  4 ,  x : x  5 a partition on the set R of real numbers.
21) Let R be the relation on the set N of positive integers defined by R = {(a,b):a + b
is even}. Is R an equivalence relation? Why or why not?
22) Show that the relation of set inclusion  is not an equivalence relation.
23) Let P be “He is tall” and let q be “he is handsome”. Write each of the following
statements in symbolic form using p and q:
a)
b)
c)
d)
He is tall and handsome
He is tall but not handsome
It is false that he is short or handsome
He is neither tall nor handsome
24) Find the truth table for (a) p  q (b) p  q .
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25) Prove the distributive law: p   q  r    p  q    p  r  using truth tables.
26) Express  in terms of  and  by showing p  q    p  q 
27) Verify that  p  q    p  q  is a tautology.
28) Determine the contrapositive of each statement (a) If John is a poet, then he is
poor. (b) Only if Marc studies will he pass the test.
29) Simplify (a) ( p  q) (b)   p  q 
30) Let A = {1,2,3,4} be the universal set. Determine the truth value of each
statement:
a) x, x  3  6
b) x, x  3  6
c) x, 2 x 2  x  15
31) Convert 45598410 into
a) Base 2
b) Base 16
32) Convert FA6547EB16 into
a) Base 10
b) Base 2
33) Find
a) DA654B16 + FE543A16
b) 1101100112 + 1001001112
34) Find a recurrence relation and initial conditions that generate a sequence
beginning with the following terms:
5, 8, 11, 14
35) Find the closed formulas for each of the following sequence.
a) 7, 13, 25, 43, 67,…
b) 5, 15, 45, 135, 405,…
c) 5, 9, 21, 57,…
36) Find a closed formula for the following recursive relation.
an  3an1  2 n  1 and a1  1
37) A bacteria culture grows at a rate of 10% per day.
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a) Today the culture has 1,000,000 bacteria. How many days will it take for
this number to double?
b) Suppose each day you remove a sample of 50,000 bacteria for testing.
How many days will it take for the original 1,000,000 bacteria to double.
38) Assume a person invests 5000 euros at 8% interest compounded annually.
An = amount in account at end of n years. A0
a) Find the recurrence relation.
b) Find a closed formula for An .
c) How long until the initial investment is tripled.
39) For the following recurrence relation, find a closed formula
an  2an1  3an2 a0  2 a1  3
40) For the following recurrence relation, find a closed formula
an  7an1  12an2 a0  3 a1  5
41) The following statement is false. Provide a counter example.
If a n b, then 2a n 2b
42) Find solutions for x in the following equation.
x 2 17 1
43) Solve 5 x 9 8
44) Prove that 42 n 10 6
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