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
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
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
how many permutations of the letters ABCDEFGH contain the string CDE
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
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
a contradiction is a proposition that is always
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
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 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
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
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
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}}
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
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
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
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
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
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
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
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!
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
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
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
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
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!
The cardinality of a finite set A={Ø,{Ø},{Ø,{Ø}}}, denoted by
3
p V ᆨq -> ᆨr is equivalent to
(p V (ᆨq)) -> ᆨr)
for proof-by-cases we use the tautology
->, <->, ^, ^
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. If n is 20 what is the probability that a person picks the correct six numbers out of?
14!6!/20!
if we randomly select 200 people at least how many of them must have birthdays on the same month?
17
how many permutations of the letters AEIOU contains the string IO
4!
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
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 {a,b,c,d,e,f,g} end with a?
6!
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)
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
in the truth table for p V ᆨq -> if p,q, r all True, then the output will be
false
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
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)
here are one example we can't apply bayes theorem. Which one is that
to do the language translation accurately.
what is the power set of an empty set
{Ø}
the cardinality of a finite set A, denoted by
|A|
