cs 1382 final exam questions
suppose that a coin is biased so that 3 appears thrice and 2 appears twice as often as each other number, but that the other 4 outcomes are equally likely. what is the probability that 5 appears when we roll this dice?
1/9
there are many lotteries that award prizes to people who correctly choose a set of six numbers out of the first n positive integers. If n is 20 what is the probability that a person picks the correct six numbers out of?
14!6!/20!
how many rows will appear on the truth table appears on the (p ∧ r ∧ t) ↔ (p ∨ q)
16
in the truth table for (p V q) -> (ᆨr V s), there will be
16
if we randomly select 200 people at least how many of them must have birthdays on the same month?
17
if U is the set of all English lowercase letters (there are 26 English letters), A = {a,e,i,o,u} B={a,b,c,d,e} are subset of U, what is the value of |A U B|
18
suppose that a die is biased so that 3 appears twice as often as each other number, but the other five outcomes are equally likely. what is the probability that 5 appears when we roll this die
1/7
suppose that a die is biased so that 3 appears thrice as often as each other number, but the other five outcomes are equally likely. what is the probability that 5 appears when we roll this die
1/8
there are many lotteries that award prizes to people who correctly choose a set of six numbers out of the first n positive integers, where n is usually between 30 and 60. what is the probability that a person picks the correct six numbers out of 50?
44!6!/50!
how many strings of three decimal digits begin with an odd digit?
500
let a @ b = max {a, b} = a if a ≥ b, otherwise a @ b = max {a, b} = b. If we want to show that for all real numbers a, b, c (a @b) @ c = a @ (b @ c) how many cases we need
6
how many permutations of the letters ABCDEFGH contain the string CDE
6!
how many permutations of {a,b,c,d,e,f,g} end with a?
6!
bag 1 contains 4 white and 6 black balls while another bag 2 containes 4 white and 3 black balls. one ball is drawn at random from one of the bags and it is found to be black. find the probability that it was drawn from bag 1
7/12
if A = {1,2,3}, what will be the cardinality of its power set P(A),
8
if we randomly select 100 people at least how many of them must have birthdays on the same month?
9
express A is a proper subset of B by set notation
A ⊂ B
the domain of f is ____ and the codomain of f is ____
A,B
which one of the following is true?
C(n,r) = C(n, n-r)
(1,3) = {x | 1 < x <= 3}
FALSE
which one of the following is true for two events A and B?
P(A)=P(A|B) P(B)+P(A|𝐵̅)𝑃(𝐵̅)
cartesian product of two sets is
a set tuples
determine the truth value of each of these statements if the domain for all variables consists of all integers.
a) ∀n(n2 ≥ 0) Ans:- True b) ∃n(n2 = 2) Ans:- false c) ∀n(n2 ≥ n) Ans:- true d) ∃n(n2 < 0) Ans:- false
two propositions are equivalent if they
always have the same truth value
suppose somebody wanted to do the indirect proof of the hypothesis below "If m and n are both perfect squares, then nm is also a perfect square." which proof strategy below will be the proof by contraposition
assume mn is not a perfect square and then from there prove that m and n both are not perfect squares
suppose somebody wanted to do the indirect proof of the hypothesis below "If m and n are both perfect squares, then nm is also a perfect square." which proof strategy below will be the proof by contraposition?
assume mn is not a perfect square and then from there prove that m and n both are not perfect squares
statements involving predicates and quantifiers are logically equivalent if and only if they have the same truth values
both a and b
"Every student in your class has taken a course in Java." what will be the negative of the expression?
both b and c
for the proof "A square of an even number is a even number" what type of proof will be good
direct proof
a contradiction is a proposition that is always
false
in the truth table for p V ᆨq -> if p,q, r all True, then the output will be
false
let Q(x,y) be the statement "x + y = 0." assume that U is the real numbers. then ∀x ∃yQ(x,y) is false, but ∃y∀xQ(x,y)istrue
false
"∃x ∀y Q(x,y) is False" it means
for every x there is a y for which Q(x,y) is False
"∃x ∀y Q(x,y) is false" it means
for every x there is a y for which Q(x,y) is false
translate the two statements into English. i) ∀x(C(x) →F(x)) and ii) ∀x( C(x) ∧ F(x)) here C(x): x is a comedian and F(x): x is funny U: all people.
i) All comedians are funny ii) Every human is a comedian and funny
translate the two statements into english ∀x(C(x) -> F(x)) and ∀x(C(x) ^ F(x)) C(x): x is a comedian and F(x): x is funny U: all people
i) all comedians are funny ii) every human is a comedian and funny
which one of the following is the perfect definition of the probability of an event?
if S is a finite nonempty sample space of equally likely outcomes, and E is an event, that is, a subset of S, then the probability of E is p(E) = |E|/|S|
which one of the following is the subtraction rule?
if a task can be done either in one of n1 ways or in one of n2 ways, then the total number of ways to do the task is n1 + n2 minus the number of ways to do the task that are common to the two different ways
what rule of inference is used in the following argument? premises: "If it is rainy, then the pool will be closed. It is rainy." "It is rainy" conclusion: "The pool is closed".
modus ponens
what rule of inference is used in the following argument? premises: "If it is rainy, then the pool will be closed. It is rainy." "It is rainy." conclusion: "The pool is closed."
modus tollens
if a flock of 20 pigeons roosts in a set of 19 pigeonholes, the pigeonhole principle stipulates that one of the pigeonhols must have:
more than 1 pigeon
P(n,r) = ?
n!/(n-r)! where 1 <= r <= n
C(n,r) = ?
n!/r!(n-r)! where 1 <= r <= n
is the function f(x) = x^2 from the set of integers to the integers to the set of integers onto?
no
Q(x,y) be the statement "x^2 + y^2 = 0". U: all the real numbers which one of the following is true
none
if f(x) = [x]a function from R -> Z. the function is
onto
the elements of cartesian products are
ordered tuples
which one of the following is a tautology
p V ᆨp
if two event A and B are independent which one of the following are true?
p(A ∩ B) = p(A)p(B)
let E and F be events with p(F) > 0. the conditional probability of E given F, denoted by P(E | F), is defined as:
p(E | F) = p(E ∩ F)/p(F)
let E and F be events with p(F) > 0. the conditional probability of E given F, denoted by P(E | F), defined as
p(E ∩ F)/p(F)
if E1 and E2 is the events in the sample space S. then which is true?
p(E1 U E2) = p(E1) + p(E2) - p(E1 ∩ E2)
which one is the bayes theorem?
p(F | E) = p(E | F)p(F)/ p(E | F)p(F) + p(E | ^F)p(^F)
the events E and F are independent if and only if
p(𝐸 ∩ 𝐹) = 𝑝(𝐸)𝑝(𝐹)
"show that pi is an irrational number" what type of proof will be good?
proof by contradiction
for the proof "If n^3 + 5 is odd then n is odd, where n is an integer." what type of proof will be easier
proof by contraposition
suppose we want to prove hypothesis below "If n is an integer and n2 is odd, then n is odd" what type of proof will be easier here?
proof by contraposition
given the conditional, p → q. which of the following is its converse
q -> p
if V is a Set of all vowels in the English alphabet: V = {a,e,i,o,u}. in which method the set is described.
roster method
a argument in propositional logic is a
sequence of propositions
an argument in propositional logic is a
sequence of propositions
to prove that P(n) is true for all positive integers n, we complete two steps: the step are
step 1: P(1) is true step 2: if P(k) is true P(k+1) is true
Let f:R→R be such that f(x) = x^100 is f invertible, and if so, what is its inverse?
the function f is not invertible because it is not one-to-one
let f: R->R be such that f(x) = x^3 is f invertible, and if so, what is its inverse?
the inverse is f(x) = 3√x
what will be the power set of set A
the set of all subsets of set A
here are one example we can't apply bayes theorem. Which one is that
to do the language translation accurately.
if f(x) = x and g(x) = x+1 find out f(g(f(x)))
x + 1
if p is true and q is false, then q -> p will be true
yes
given the following set builder representation of a set: M={x ∈ Z | |x| < 3}. which of the following sets satisfies this representation?
{-2, -1, 0, 1, 2}
given the following set builder representation of a set: M = {x ∈ Z | |x| < 4} which of the following sets satisfies this representation
{-3,-2,-1,0,1,2,3}
the g^-1({0}) for function g(x) = |x| is... hint: [] = floor symbol
{x | 0 <= x <}
what will be the power of set A={a,b}
{Ø,{a},{b},{a,b}}
what is the power set of an empty set
{Ø}
the cardinality of a finite set A, denoted by
|A|
which rule is the principle of inclusion-exclusion
|𝐴 ∪ 𝐵| = |𝐴| + |𝐵| − |𝐴 ∩ 𝐵|
which one of the following equivalences is a De Morgans law
ᆨ(p V q) ≡ ᆨp ^ ᆨq
which of the following correctly represents De Morgan's law of quantifiers?
ᆨ∃xP(x) ≡ ∀x ᆨP(x)
if A is a subset of B, which of the following statements is true?
∀x(x ∈ A -> x ∈ B)
If sets A and B are equal, then which of the following statements is true?
∀x(x ∈ A <-> x ∈ B)
if U consists of the integers 1,10, and 100
∀xP(x) = P(1) ^ P(10) ^ P(100)
let Q (x, y, z) be the statement "x + y = z." what are the truth values of the statements ∀x∀y∃zQ(x, y, z) and ∃z∀x∀yQ(x, y, z), where U:all the real numbers?
∀x∀y∃zQ(x, y, z): T ∃z∀x∀yQ(x, y, z): F
p V ᆨq -> ᆨr is equivalent to
(p V (ᆨq)) -> ᆨr)
which one of the following equivalence laws is wrong?
(p ^ (q V r)) ≡ (p ^ q) V ( p V r)
which of the tautology is for modus tollens
(¬𝑞 ∧ (𝑝 → 𝑞)) → ¬𝑝
for proof-by-cases we use the tautology
->, <->, ^, ^
if 𝑃(𝑛, 𝑟) = 𝑛!/(𝑛−𝑟)! then which one of the following must be true?
1 <= r <= n
what is the probability that a positive integer selected at random from the set positive integers not exceeding 100 is divisible by both 2 and 5?
1/10
what is the probability that a positive integer selected at random from the set of positive not exceeding 200 is divisible by both 2 and 5
1/20
let's say that you are conducting a survey to find out how many people in a certain city own a car and how many of them also own a bike. you ask 500 people in the city, and you find that 300 of them own a car, and 200 of them own both a car and a bike. what is the conditional probability of someone owning a bike given that they already own a car ?
2/3
there are many lotteries that award prizes to people who correctly choose a set of six numbers out of the first n positive integers, where n is usually between 30 and 60. what is the probability that a person picks the correct six numbers out of 30?
24!6!/30!
how many bit strings of length n both begin and end with 1?
2^(n-2)
how many bit strings of length 7 either start with a 11 bit
2^5
The cardinality of a finite set A={Ø,{Ø},{Ø,{Ø}}}, denoted by
3
what is the probability that a positive integer selected at random from the set of positive integers not exceeding 100 is divisible by either 2 or 5 or both?
3/5
what is the conditional probability that a randomly generated bit string of length 4 contains at least two consecutives 0s, given that the first bit is a 1? (the probabilites of 0 and 1 are same)
3/8
there are many lotteries that award prizes to people who correctly choose a set of six numbers out of the first n positive integers, where n is usually between 30 and 60. what is the probability that a person picks the correct six numbers out of 40?
34!6!/40!
how many permutations of the letters AEIOU contains the string IO
4!
a group consists of 4 girls and 7 boys. in how many ways can a team of 5 members be selected if the team has at least three girls
4!/3! X 7!/5!x2! X 4i/4i X 7i/6ix1i
let's say that you are conducting a survey to find out how many people in a certain city own a car and how many of them also own a bike. you ask 1000 people in the city, and you find that 600 of them own a car, and 400 of them own both a car and a bike. what is the conditional probability of someone owning a bike given that they already own a car ?
400/600
