Download refer these types of questions and answers and more Exams Computer Science in PDF only on Docsity!
Basic Math question answers
SETS
Georg Cantor (1845-1918), a German mathematician, initiated the concept ‘Theory of sets’ or ‘Set Theory’. While working on “ Problems on Trigonometric Series ”, He encountered sets, that have become one of the most fundamental concepts in mathematics. Without understanding sets, it will be difficult to explain the other concepts such as relations, functions, sequences, probability, geometry, etc. A set is defined as a collection of distinct objects of the same type or class of objects. The purposes of a set are called elements or members of the set. An object can be numbers, alphabets, names, etc. A Set is an unordered collection of objects, known as elements or members of the set. An element ‘a’ belong to a set A can be written as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A. Examples of sets are: (a) A set of rivers of India. (b) A set of vowels. We broadly denote a set by the capital letter A, B, C, etc. while the fundamentals of the set by small letter a, b, x, y, etc. If A is a set, and a is one of the elements of A, then we denote it as a ∈ A. Here the symbol ∈ means -"Element of." Sets Representation Sets are represented in two forms:- a) Statement form. b) Roster or tabular form: In this form of representation we list all the elements of the set within braces { } and separate them by commas. Example: If A= set of all odd numbers less then 10 then in the roster from it can be expressed as A={ 1,3,5,7,9}. c) Set Builder form: In this form of representation we list the properties fulfilled by all the elements of the set. We note as {x: x satisfies properties P}. and read as 'the set of those entire x such that each x has properties P.' Example: If B= {2, 4, 8, 16, 32}, then the set builder representation will be: B={x: x= n , where n ∈ N and 1≤ n ≥5} Statement form In this representation, the well-defined description of the elements of the set is given. Below are some examples of the same.
- The set of all even number less than 10.
- The set of the number less than 10 and more than 1. Roster form COPYRIGHT DIWAKAR EDUCATION HUB Page 1 BASIC MATHEMATICS UNIT- 1 In this representation, elements are listed within the pair of brackets {} and are separated by commas. Below are two examples.
- Let N is the set of natural numbers less than 5. N = { 1 , 2 , 3, 4 }.
- The set of all vowels in the English alphabet. V = { a , e , i , o , u }. Set builder form In Set-builder set is described by a property that its member must satisfy.
- {x: x is even number divisible by 6 and less than 100}.
- {x: x is natural number less than 10}. Standard Notations: x ∈ A x belongs to A or x is an element of set A. x ∉ A x does not belong to set A. ∅ Empty Set. U Universal Set. N The set of all natural numbers. I The set of all integers. I 0 The set of all non- zero integers. I+ The set of all + ve integers. C, C 0 The set of all complex, non-zero complex numbers respectively. Q, Q 0 , Q+ The sets of rational, non- zero rational, +ve rational numbers respectively. R, R 0 , R+ The set of real, non-zero real, +ve real number respectively. Size of a Set Size of a set can be finite or infinite. For example
3. Subsets: If every element in a set A is also an element of a set B, then A is called a subset of B. It can be denoted as A ⊆ B. Here B is called Superset of A. To prove A is the subset of B, we need to simply show that if x belongs to A then x also belongs to B. To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B. ‘U’ denotes the universal set. Above Venn Diagram shows that A is a subset of B. Example: If A= {1, 2} and B= {4, 2, 1} the A is the subset of B or A ⊆ B. Properties of Subsets: COPYRIGHT DIWAKAR EDUCATION HUB Page 3 BASIC MATHEMATICS UNIT- 1
- Every set is a subset of itself.
- The Null Set i.e.∅ is a subset of every set.
- If A is a subset of B and B is a subset of C, then A will be the subset of C. If A⊂B and B⊂ C ⟹ A ⊂ C
- A finite set having n elements has 2 n subsets. 4. Proper Subset: If A is a subset of B and A ≠ B then A is said to be a proper subset of B. If A is a proper subset of B then B is not a subset of A, i.e., there is at least one element in B which is not in A. Example: (i) Let A = {2, 3, 4} B = {2, 3, 4, 5} A is a proper subset of B. (ii) The null ∅ is a proper subset of every set. 5. Improper Subset: If A is a subset of B and A = B, then A is said to be an improper subset of B. Example (i) A = {2, 3, 4}, B = {2, 3, 4} A is an improper subset of B. (ii) Every set is an improper subset of itself.
6. Universal Set: If all the sets under investigations are subsets of a fixed set U, then the set U is called Universal Set. Example: In the human population studies the universal set consists of all the people in the world. 7. Null Set or Empty Set: A set having no elements is called a Null set or void set. It is denoted by∅. 8. Singleton Set: It contains only one element. It is denoted by {s}. Example: S= {x|x∈N, 7<x<9} = {8} 9. Equal Sets: Two sets A and B are said to be equal and written as A = B if both have the same elements. Therefore, every element which belongs to A is also an element of the set B and every element which belongs to the set B is also an element of the set A. A = B ⟺ {x ϵ A ⟺ x ϵ B}. If there is some element in set A that does not belong to set B or vice versa then A ≠ B, i.e., A is not equal to B. NOTE: Order of elements of a set doesn’t matter. 10. Equivalent Sets: If the cardinalities of two sets are equal, they are called equivalent sets. Example: If A= {1, 2, 6} and B= {16, 17, 22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A|=|B|= 11. Disjoint Sets: Two sets A and B are said to be disjoint if no element of A is in B and no element of B is in A. Example: R = {a, b, c} COPYRIGHT DIWAKAR EDUCATION HUB Page 4 BASIC MATHEMATICS UNIT- 1 S = {k, p, m} R and S are disjoint sets. 12. Power Sets: The power of any given set A is the set of all subsets of A and is denoted by P (A). If A has n elements, then P (A) has 2 n elements. Example: A = {1, 2, 3} P (A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. Partitions of a Set: Let S be a nonempty set. A partition of S is a subdivision of S into nonoverlapping, nonempty subsets. Speceficially, a partition of S is a collection {Ai} of nonempty subsets of S such that: o Each a in S belongs to one of the Ai. o The sets of {Ai} are mutually disjoint; that is,
was suggested by John Venn. He represented the relationship between different groups of things in the pictorial form that is known as a Venn diagram. Examples: A set is a collection or group of things. It may contain digits, vowels, animals, prime numbers, etc. A set is denoted by the capital letter and the elements of the set are denoted by the lowercase letters. All the elements of a set enclose in the pair of curly braces {}. For example, E is a set that denotes even numbers less than 10. We can represent it in the form of set, as follows: E = {2,4,6,8} Where E is the set name and 2, 4, 6, 8 are the elements of the set. We can also represent a set in pictorial form that is known as Venn diagram. What is the Venn diagram? A diagram or figure that represents the mathematical logic or relation between a finite collection of different sets (a group of things) is called the Venn diagram. It is used to illustrate the set relationship. We usually use a circle or oval to represent a Venn diagram. It may have more than one circle; each represents a set. Suppose there are two sets A and B having the elements {1, 2, 3} and {8, 5, 9}, respectively. We can represent these two sets in the Venn diagram, as shown below. Advantages of Venn Diagram o It is used for both comparison and classification. COPYRIGHT DIWAKAR EDUCATION HUB Page 6
BASIC MATHEMATICS UNIT- 1
o It groups the information into different parts. o It also highlights the similarities and differences. Uses of Venn Diagram Venn diagram is used in mathematics to understand the set theory. We also use it to understand the relationship between or among sets of objects. It depicts the set of intersections and unions. How to Draw a Venn Diagram o First, we draw a rectangle. o Write the union () sign either the left or right top corner of the rectangle. o Inside the rectangle, write the elements that do not belong to any set. Draw circle(s) that represent the set. o Inside or outside of the circle, write the name of the corresponding set. o Inside the circle, write the elements of the set. Suppose there are two sets, A and B, having some elements in common. The Venn diagram of the sets can be drawn as follows: Types of Venn Diagram There are following types of Venn diagram: o Two-Set Diagrams o Two-Set Euler Diagrams o Three-Set Diagrams o Three-Set Euler Diagrams o Four-Set Diagrams Two-Set Diagrams: When two sets overlap each other is called two-set diagrams. In the following diagram, there are two sets, A and B , having the elements {a, e, i, o, u} and {a, b, c, d,
COPYRIGHT DIWAKAR EDUCATION HUB Page 8 BASIC MATHEMATICS UNIT- 1 Three-Set Diagrams: When three sets overlap, each other is called three-set diagrams. In the following diagram, there are three sets A, B, and C having the elements {Andrew, Peter, Sam, Tom, David}, {Michael, Sam, Tom, Robert, Jack, Smith}, and {David, Maria, Tom, Angelina, Michael}, respectively. Set A, B, and C represents the name of students who learn physics, chemistry, and math, respectively. In the above sets, some students learn only one subject, some students learn tow subjects, and some students learn all three subjects. o David is the student who learns both physics and math. o Michael is the student who learns both chemistry and math. o Sam is the student who learns both chemistry and physics. o Tom is the student who learns three-subject chemistry, math, and physics. Therefore, we can represent it in the Venn diagram, as shown below. Three-Set Euler Diagram: In the three-set diagram, when a set does not overlap the other two sets is called a three-set Euler diagram. Suppose, the set A represents the set of vowels {a, e, i,
o, u}, set B represents the set of alphabets {a, b, c, d, e, f, g}, and set C represents the set of Greek letters {ρ, ω, φ, θ, ϵ}. We can represent it in the Venn diagram, as follows. The three-set Euler diagram can have a nested set. In the following diagram, the pink things set may contain a set of light-pink things. COPYRIGHT DIWAKAR EDUCATION HUB Page 9 BASIC MATHEMATICS UNIT- 1 The above diagram is not a Venn diagram because two sets do not overlap (Black Things & light-pink Things) each other. Four-Set Diagram: We use an oval shape to represent the four-set diagram because the circle no longer overlaps each other. The oval shape ensures that all sets overlap each other. It is the only option to represent the four-set diagram.
Answer: d Explanation: Set = {0} non-empty and finite set.
- {x: x is a real number between 1 and 2} is an ________ a) Infinite set b) Finite set c) Empty set d) None of the mentioned Answer: a Explanation: It is an infinite set as there are infinitely many real number between any two different real numbers.
- Write set {1, 5, 15, 25,…} in set-builder form. a) {x: either x=1 or x=5n, where n is a real number} b) {x: either x=1 or x=5n, where n is a integer} c) {x: either x=1 or x=5n, where n is an odd natural number} d) {x: x=5n, where n is a natural number} Answer: c Explanation: Set should include 1 or an odd multiple of 5.
- Express {x: x= n/ (n+1), n is a natural number less than 7} in roster form. a) { 1 ⁄ 2 , 2 ⁄ 3 , 4 ⁄ 5 , 6 ⁄ 7 } b) { 1 ⁄ 2 , 2 ⁄ 3 , 3 ⁄ 4 , 4 ⁄ 5 , 5 ⁄ 6 , 6 ⁄ 7 , 7 ⁄ 8 } c) { 1 ⁄ 2 , 2 ⁄ 3 , 3 ⁄ 4 , 4 ⁄ 5 , 5 ⁄ 6 , 6 ⁄ 7 } d) Infinite set Answer: c Explanation: n/(n+1) = 1/(1+1) = 1 ⁄ 2 and n>7.
- Number of power set of {a, b}, where a and b are distinct elements. a) 3 b) 4 c) 2 d) 5 Answer: b Explanation: Power set of {a, b} = {∅, {a, b}, {a}, {b}}.
- Which of the following is subset of set {1, 2, 3, 4}? a) {1, 2} b) {1, 2, 3} c) {1} d) All of the mentioned Answer: d Explanation: There are total 16 subsets.
- A = {∅,{∅},2,{2,∅},3}, which of the following is true? a) {{∅,{∅}} ∈ A b) {2} ∈ A c) ∅ ⊂ A d) 3 ⊂ A Answer: c Explanation: Empty set is a subset of every set.
- Subset of the set A= { } is? a) A b) {} c) ∅ d) All of the mentioned COPYRIGHT DIWAKAR EDUCATION HUB Page 153 BASIC MATHEMATICS UNIT- 1 Answer: d Explanation: Every set is subset of itself and Empty set is subset of each set.
- {x: x ∈ N and x is prime} then it is
________
a) Infinite set b) Finite set c) Empty set
d) Not a set Answer: a Explanation: There is no extreme prime, number of primes is infinite.
- Convert set {x: x is a positive prime number which divides 72} in roster form. a) {2, 3, 5} b) {2, 3, 6} c) {2, 3} d) {∅} Answer: c Explanation: 2 and 3 are the divisors of 72 which are prime.
- A __________ is an ordered collection of objects. a) Relation b) Function c) Set d) Proposition Answer: c Explanation: By the definition of set. 12. The set O of odd positive integers less than 10 can be expressed by
a) {1, 2, 3} b) {1, 3, 5, 7, 9} c) {1, 2, 5, 9} d) {1, 5, 7, 9, 11} Answer: b Explanation: Odd numbers less than 10 is {1, 3, 5, 7, 9}.
- Power set of empty set has exactly _________ subset. a) One b) Two c) Zero d) Three Answer: a Explanation: Power set of null set has exactly one subset which is empty set. 14. What is the Cartesian product of A = {1, 2} and B = {a, b}? a) {(1, a), (1, b), (2, a), (b, b)} b) {(1, 1), (2, 2), (a, a), (b, b)} c) {(1, a), (2, a), (1, b), (2, b)} d) {(1, 1), (a, a), (2, a), (1, b)} Answer: c Explanation: A subset R of the Cartesian product A x B is a relation from the set A to the set B.
- The Cartesian Product B x A is equal to the Cartesian product A x B. a) True b) False Answer: b Explanation: Let A = {1, 2} and B = {a, b}. The Cartesian product A x B = {(1, a), (1, b), (2, a), (2, b)} and the Cartesian product B x A = {(a, 1), (a, 2), (b, 1), (b, 2)}. This is not equal to A x B.
- What is the cardinality of the set of odd positive integers less than 10? a) 10 b) 5 c) 3 d) 20 Answer: b Explanation: Set S of odd positive an odd integer less than 10 is {1, 3, 5, 7, 9}. Then, Cardinality of set S = |S| which is 5. 17. Which of the following two sets are equal? a) A = {1, 2} and B = {1} b) A = {1, 2} and B = {1, 2, 3} c) A = {1, 2, 3} and B = {2, 1, 3} d) A = {1, 2, 4} and B = {1, 2, 3} COPYRIGHT DIWAKAR EDUCATION HUB Page 154
and B denoted by A-B, is the set containing those elements that are in A not in B. 26. The complement of the set A is
COPYRIGHT DIWAKAR EDUCATION HUB Page 155 BASIC MATHEMATICS UNIT- 1 a) A – B b) U – A c) A – U d) B – A Answer: b Explanation: The complement of the set A is the complement of A with respect to U. 27. The bit string for the set {2, 4, 6, 8, 10} (with universal set of natural numbers less than or equal to 10) is
a) 0101010101 b) 1010101010 c) 1010010101 d) 0010010101 Answer: a Explanation: The bit string for the set has a one bit in second, fourth, sixth, eighth, tenth positions, and a zero elsewhere. 28. Let Ai = {i, i+1, i+2, …..}. Then set {n, n+1, n+2, n+3, …..} is the _________ of the set Ai. a) Union b) Intersection c) Set Difference d) Disjoint Answer: b Explanation: By the definition of the generalized intersection of the set. 29. The bit strings for the sets are 1111100000 and
- The union of these sets is
a) 1010100000 b) 1010101101 c) 1111111100 d) 1111101010 Answer: d Explanation: The bit string for the union is the bitwise OR of the bit strings. 30. The set difference of the set A with null set is __________ a) A b) null c) U d) B
Answer: a Explanation: The set difference of the set A by the null set denoted by A – {null} is A. 31. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then the number of elements in A U B is? a) 4 b) 5 c) 6 d) 7 Answer: a Explanation: AUB is {1, 2, 3, 4}.
- Let the set A is {1, 2, 3} and B is { 2, 3, 4}. Then number of elements in A ∩ B is? a) 1 b) 2 c) 3 d) 4 Answer: b Explanation: A ∩ B is {2, 3}.
- Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then the set A – B is? a) {1, -4} b) {1, 2, 3} c) {1} d) {2, 3} Answer: c
Explanation: In A – B the common elements get cancelled.
- In which of the following sets A – B is equal to B – A? a) A = {1, 2, 3}, B = {2, 3, 4} b) A = {1, 2, 3}, B = {1, 2, 3, 4} c) A = {1, 2, 3}, B = {2, 3, 1} d) A = {1, 2, 3, 4, 5, 6}, B = {2, 3, 4, 5, 1} Answer: c Explanation: A- B= B-A = Empty set. 35. Let A be set of all prime numbers, B be the set of all even prime numbers, C be the set of all odd prime numbers, then which of COPYRIGHT DIWAKAR EDUCATION HUB Page 156 BASIC MATHEMATICS UNIT- 1 the following is true? a) A ≡ B U C b) B is a singleton set. c) A ≡ C U {2} d) All of the mentioned Answer: d Explanation: 2 is the only even prime number.
- If A has 4 elements B has 8 elements then the minimum and maximum number of elements in A U B are ____________ a) 4, 8 b) 8, 12 c) 4, 12 d) None of the mentioned Answer: b Explanation: Minimum would be when 4 elements are same as in 8, maximum would be when all are distinct.
- If A is {{Φ}, {Φ, {Φ}}}, then the power set of A has how many element? a) 2 b) 4 c) 6 d) 8 Answer: b Explanation: The set A has got 2 elements so n(P(A))=4.
- Two sets A and B contains a and b elements respectively. If power set of A contains 16 more elements than that of B, value of ‘b’ and ‘a’ are _______ a) 4, 5 b) 6, 7 c) 2, 3 d) None of the mentioned Answer: a Explanation: 32-16=16, hence a=5, b=4. 39. Let A be {1, 2, 3, 4}, U be set of all natural numbers, then U-A’(complement of A) is given by set. a) {1, 2, 3, 4, 5, 6, ….} b) {5, 6, 7, 8, 9, ……} c) {1, 2, 3, 4} d) All of the mentioned Answer: c Explanation: U – A’ ≡ A.
- Which sets are not empty? a) {x: x is a even prime greater than 3} b) {x : x is a multiple of 2 and is odd} c) {x: x is an even number and x+3 is even} d) { x: x is a prime number less than 5 and is odd} Answer: d Explanation: Because the set is {3}.
- The shaded area of figure is best described by? a) A ∩ B
will give the result 1, i.e., B.0 = 0 B+1 = 1 Identity Law When the variable is AND with 1 and OR with 0, the variable remains the same, i.e., B.1 = B B+0 = B Idempotent Law When the variable is AND and OR with itself, the variable remains same or unchanged, i.e., B.B = B B+B = B Complement Law When the variable is AND and OR with its complement, it will give the result 0 and 1 respectively. B.B' = 0 B+B' = 1 Double Negation Law COPYRIGHT DIWAKAR EDUCATION HUB Page 1 DIGITAL LOGIC UNIT- 2 This law states that, when the variable comes with two negations, the symbol gets removed and the original variable is obtained. ((A)')' = A Commutative Law This law states that no matter in which order we use the variables. It means that the order of variables doesn't matter in this law. A.B = B.A A+B = B+A Associative Law This law states that the operation can be performed in any order when the variables priority is of same as '*' and '/'. (A.B).C = A.(B.C) (A+B)+C = A+(B+C) Distributive Law This law allows us to open up of brackets. Simply, we can open the brackets in the Boolean expressions. A+(B.C) = (A+B).(A+C
A.(B+C) = (A.B)+(A.C)
Absorption Law This law allows us for absorbing the similar variables. B+(B.A) = B B.(B+A) = B De Morgan Law The operation of an OR and AND logic circuit will remain same if we invert all the inputs, change operators from AND to OR and OR to AND, and invert the output. (A.B)' = A'+B' (A+B)' = A'.B' Sub-Algebra Consider a Boolean-Algebra (B, , +,', 0,1) and let A ⊆ B. Then (A,, +,', 0,1) is called a sub algebra or Sub-Boolean Algebra of B if A itself is a Boolean Algebra i.e., A contains the elements 0 and 1 and is closed under the operations , + and '. Example: Consider the Boolean algebra D 70 whose Hasse diagram is shown in fig: COPYRIGHT DIWAKAR EDUCATION HUB Page 2 DIGITAL LOGIC UNIT- 2 Clearly, A= {1, 7, 10, 70} and B = {1, 2, 35, 70} is a sub-algebra of D 70. Since both A and B are closed under operation ∧,∨and '. Note: A subset of a Boolean Algebra can be a Boolean algebra, but it may or may not be sub algebra as it may not close the operation on B. Isomorphic-Boolean Algebras: Two Boolean algebras B and B 1 are called isomorphic if there is a one to one correspondence f: B⟶B 1 which preserves the three operations +, and ' for any elements a, b in B i.e., f (a+b)=f(a)+f(b) f (ab)=f(a)f(b) and f(a')=f(a)'. Example: The following are two distinct Boolean algebras with two elements which are isomorphic. 1.The first one is a Boolean Algebra that is derived from a power set P(S) under ⊆
