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WorksheetsCIT 201 - Discrete Structures
Total questions: 85
Worksheet time: 41mins
What is the minimum number of cells in a three-variable K-map?
4
6
8
16
How many cells are in a four-variable K-map?
8
12
16
24
What is the purpose of "Don't Care" conditions in Karnaugh maps?
To minimize literals in the solution
To represent undecidable outputs
To find essential prime implicants
To increase complexity
How many adjacent cells are needed to form a group that eliminates three variables in a K-map?
2
4
8
16
Which is NOT a valid grouping rule for K-maps?
Groups must be in powers of two
Groups can overlap
Groups must always be rectangular
Groups can include diagonals
If a password consists of 4 letters followed by 2 digits, how many possible passwords can be made?
26*4*10^2
26*4*10*2
26*10*4
26*4+10^2
What is the value of 5P3 (permutation of 5 items taken 3 at a time)?
20
60
120
10
How many ways can a committee of 4 be selected from 10 people?
10P4
10C4
4C10
10*4
Which principle ensures that if 10 objects are placed into 9 boxes, at least one box contains more than one object?
Basic Counting Principle
Permutation Rule
Pigeonhole Principle
Combination Rule
A license plate consists of 3 letters and 3 digits. How many unique plates are possible?
3*26*10^3
26*6+10^6
3*26*10
26*3*10^2
How many ways can the letters in the word PROBABILITY be arranged?
11!
11!/3!
11!/2!
11!/2!*2!*3!
What is the value of 8C2?
16
28
56
112
Which formula calculates the number of ways to choose k objects from n objects?
n!/(n−k)!
n!/(k!(n−k)!)
k!/(n−k)!
n!/(n−k)
What is the probability of flipping a coin twice and getting at least one head?
0.25
0.5
0.75
1
If two events A and B are independent, P(A∩B) is:
P(A)+P(B)
P(A)⋅P(B)
P(A)/P(B)
P(A)−P(B)
What is the complement of P(A)?
1−P(A)
P(Ac)
Both a and b
None of the above
Given P(A|B)=0.4 and P(B)=0.5, find P(A∩B):
0.2
0.5
0.1
0.4
What is the probability of drawing a red card or a queen from a standard deck of cards?
13/52
27/52
26/52
28/52
The Cartesian product A×B contains 15 elements. If A has 3 elements, how many elements does B have?
3
5
15
45
A relation that is symmetric, transitive, and reflexive is called:
Partial relation
Equivalence relation
Asymmetric relation
Irreflexive relation
If R is a relation on a set A={1,2,3}, how many subsets does R have?
8
64
512
27
A relation is defined as R={(x,y)∣x≥y}. Which property does R satisfy?
Reflexive and symmetric
Reflexive and transitive
Symmetric and transitive
None of the above
Which property is violated if aRb and bRa, but a≠b?
Reflexivity
Symmetry
Transitivity
Anti-symmetry
How many adjacent cells does a single cell in a four-variable K-map have?
2
3
4
5
What is the largest group of cells that can be formed in a four-variable K-map?
4
8
12
16
Which type of K-map group eliminates two variables in a simplified expression?
Pair
Quad
Octet
Single
In a three-variable K-map, a group of 4 adjacent cells eliminates:
One variable
Two variables
Three variables
No variables
How many groups can be formed from "Don't Care" conditions in a three-variable K-map?
1
2
Multiple groups
No groups
How many 3-digit numbers can be formed using the digits 1 to 9 without repetition?
504
729
81
1000
How many 3-digit numbers can be formed using the digits 1 to 9 without repetition?
504
729
81
1000
A lock has a 4-digit code, and digits can repeat. How many possible codes exist?
10^4
10!/4!
4⋅10
10!/(10−4)!
What is the total number of outcomes when rolling three six-sided dice?
6
18
216
1296
In how many ways can 7 people sit in a circle?
7!
6!
7⋅6!
7!/7
If P(n,k)=20 and k=2, find n:
5
4
3
6
How many subsets does a set with 7 elements have?
49
64
128
256
In how many ways can 4 men and 4 women be arranged such that no two women are adjacent?
4!⋅4!
4!⋅5!
4!2
4!⋅6!
How many ways can a 10-question test be answered if each question has 4 options?
10^4
4^10
10⋅4
10!/4!
What is the probability of rolling a sum of 7 with two six-sided dice?
1/12
1/6
1/9
1/36
What is the probability of selecting a king or a heart from a standard deck of cards?
13/52
16/52
15/52
17/52
What is the probability of drawing two aces consecutively from a standard deck without replacement?
4/52⋅3/52
4/52⋅3/51
1/169
4!/2!
A coin is tossed 3 times. What is the probability of getting exactly 2 heads?
1/8
3/8
1/2
7/8
If P(A)=0.6 and P(Ac)=0.4, what is P(A∪Ac)?
0
0.4
0.6
1
How many elements are in the Cartesian product of two sets with 4 and 6 elements, respectively?
24
10
30
40
What is the total number of relations between two sets with 3 and 2 elements, respectively?
6
64
36
8
A relation R on set A is said to be antisymmetric if:
aRb implies bRa
aRb and bRa imply a=b
aRa is true for all a∈A
aRb and bRa imply a≠b
What property does the relation R={(x,x)∣x∈A} satisfy?
Reflexive only
Reflexive and symmetric
Symmetric and transitive
Reflexive, symmetric, and transitive
Which property is violated by the relation R={(a,b),(b,a)} where a≠b?
Reflexivity
Symmetry
Transitivity
Antisymmetry
How many equivalence relations exist on a set with 3 elements?
3
4
5
6
What is the maximum number of ordered pairs in a relation on a set with 5 elements?
10
15
25
50
A binary relation R that is both reflexive and antisymmetric is called:
A partial order
A total order
An equivalence relation
A function
What is the main advantage of using a K-map for simplification?
Eliminates the need for Boolean algebra
Ensures maximum literals in the expression
Visually identifies redundancies
Avoids the need for truth tables
If a four-variable K-map group covers four adjacent cells, what simplification occurs?
Two variables are eliminated
One variable is eliminated
No variables are eliminated
All variables are eliminated
If a bookshelf contains 5 mathematics books and 3 physics books, how many ways can the books be arranged such that all mathematics books are together?
6!⋅5!
5!⋅4!
3!⋅5!
5!⋅3!
In a lottery, 6 numbers are drawn from a set of 50. How many different combinations of numbers are possible?
50P6
50C6
50!/6!
50!/(44!)
A bag contains 4 red balls, 3 blue balls, and 2 green balls. What is the probability of randomly selecting a blue ball?
1/3
3/9
3/4
1/4
A spinner is divided into 5 equal sectors numbered 1 through 5. What is the probability of landing on an odd number?
1/5
2/5
3/5
4/5
A relation is a function if:
Every element in the domain maps to a unique element in the codomain
Every element in the codomain maps to the domain
The relation is reflexive and symmetric
The relation is transitive and antisymmetric
Which Boolean law does K-Map grouping rely on to eliminate variables?
(a)
What counting principle states that if one task can be done in m ways and another task in n ways, the total number of ways is m x n?
(a)
What is the name of the counting principle that deals with dividing objects into groups where the order does not matter?
(a)
What is the result of 4!?
(a)
Which principle states that if n + 1 objects are placed into n boxes, at least one box must contain more than one object?
(a)
What is the term for a function that maps each element of a set to exactly one element of another set?
(a)
If there are 3 shirts and 2 pants, how many unique outfits can be made?
(a)
If a password consists of 3 letters followed by 2 digits, how many unique passwords are possible if repetition is allowed?
(a)
If two events cannot happen at the same time, what are they called?
(a)
What is a relation that is symmetric but not transitive?
(a)
The largest group that can be formed in a three-variable K-map is:
2 cells
4 cells
6 cells
8 cells
Don’t-care conditions in a K-map are used to:
Ignore certain minterms
Eliminate invalid outputs
Simplify expressions
Ensure all variables are used
A combination lock has 3 dials, each with 10 digits. How many possible outcomes exist?
300
500
1000
100
Evaluate P (6 , 3):
20
60
120
720
If two dice are rolled, what is the probability that the sum is 7?
0.0833
0.1666
0.125
0.027777
Find the Cartesian product 𝐴 × 𝐵 A×B where 𝐴 = { 1 , 2 } A={1,2} and 𝐵 = { 𝑎 , 𝑏 } B={a,b}.
{1a,2b}
{(1,a),(2,b)}
{(1,a),(1,b),(2,a),(2,b)}
{(1,1),(2,2)}
A relation is symmetric if:
(𝑎,𝑏)∈𝑅⟹(𝑏,𝑎)∉𝑅(a,b)∈R⟹(b,a)∈/R
(a,b)∈R⟹(b,a)∈R
(a,a)∈R∀a∈A
None of these
Is the relation 𝑅 = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 1 , 2 ) , ( 2 , 1 ) } R={(1,1),(2,2),(3,3),(1,2),(2,1)} an equivalence relation on 𝐴 = { 1 , 2 , 3 } A={1,2,3}?
Yes
No
Not enough information
None of these
What is the process of grouping adjacent 1s in a K-map to simplify a Boolean expression?
(a)
What is the likelihood of an event happening, expressed as a fraction of favorable outcomes over total outcomes?
(a)
What is the set of all ordered pairs formed by taking elements from two sets?
(a)
What type of relation satisfies reflexivity, symmetry, and transitivity?
(a)
If a relation is not reflexive, symmetric, or transitive, it is typically classified as what?
(a)
A drawer contains 15 pairs of socks in 3 different colors (red, blue, and green). If you randomly pick socks without looking, how many socks must you pick to guarantee having at least two matching pairs of the same color?
7
6
8
9
In a group of 100 people, each person shakes hands with at least 10 other people. How many people must you select to ensure that at least two people have shaken hands with the same set of individuals?
11
10
14
16
Which of the following relations is transitive?
If ( 1 , 2 ) ∈ 𝑅 (1,2)∈R and ( 2 , 3 ) ∈ 𝑅 (2,3)∈R, then ( 1 , 3 ) ∈ 𝑅 (1,3)∈R.
If ( 1 , 2 ) ∈ 𝑅 (1,2)∈R and ( 2 , 3 ) ∈ 𝑅 (2,3)∈R, then ( 1 , 2 ) ∈ 𝑅 (1,2)∈R.
If ( 1 , 2 ) ∈ 𝑅 (1,2)∈R and ( 3 , 4 ) ∈ 𝑅 (3,4)∈R, then ( 1 , 4 ) ∈ 𝑅 (1,4)∈R.
If ( 1 , 1 ) ∈ 𝑅 (1,1)∈R and ( 1 , 2 ) ∈ 𝑅 (1,2)∈R, then ( 2 , 2 ) ∈ 𝑅 (2,2)∈R.
In how many ways can 4 books be arranged on a shelf if 2 specific books must not be placed next to each other?
12
20
18
30
How many ways are there to select a team of 5 players from a group of 12 players if the team must consist of at least 2 women, given that there are 8 men and 4 women in the group?
336
252
420
560
