Chapter 2

Question 1

Well defined collection def

Answer 1

Well-defined collection is meant a collection, which is such that given any object, we may be able to decide whether the object belongs to the collection or not

Question 2

Distinct objects

Answer 2

By distinct objects, we mean objects no two of which are identical or same

Question 3

Ways of describing a set

Answer 3

Discriptive
Tabular
Set builder

Question 4

Discriptive method def

Answer 4

A set may be described in words. For instance, the set of all the vovals of the English alphabets

Question 5

Tabular method def

Answer 5

A set may be described by listingits elements within brackets. If set A is the set of english vowels then
A={a,e,i,o,u}

Question 6

Set builder method

Answer 6

This is done by using a symbol or letter for an arbitrary member of the set and stating the property common to all numbers. Thus, a set of vowels may be written as
A={x|x is a vowel of the english alphabetsl}

Question 7

Equal set def

Answer 7

Two sets A and B are equal, i.e., A=B if and only if they have the same elements, that is, if and only if every element of each is an element of another set.
Thus, A={3,2,1} and {1,2,3} are equal

Question 8

Equivalent set def

Answer 8

If elements of two sets A and B can be paired in such a way that each element of A is paired with one and only one element of B and vice versa. Then, such a pairing is called one to one correspondence between A and B.

E.g,
A ={ali, ahmed, bilal}
B={fatima, ayesha, samina}

Question 9

Singleton set def

Answer 9

A set having only one element is called singleton set.

Question 10

Empty set def

Answer 10

A set with no elements (zero number of elements) is called empty or null set.
Denoted by Ö or {}.

Question 11

Finite set def

Answer 11

If a set is equivalent to the set {1,2,3,….n} for some fixed natural number n then set is called finite.
N={1,2,3,……..99}

Question 12

Infinite set def

Answer 12

If a set is not equivalent to the set {1,2,3,….n} for some fixed natural number n, then set is called infinite.
N={1,2,3,……..}

Question 13

Subset def

Answer 13

If every element of set A is an element of set B, then A is a subset of B.
Symbiotically, it is written as A c_ B

Question 14

Proper subset def

Answer 14

If A is a subset of B and B contains at least one element, which is not an element of A. Then, A is said to be a proper subset of B.

Question 15

Improper subset

Answer 15

If A is a subset of B and A=B, then we say that A is an improper subset of B.

Question 16

Power set def

Answer 16

The power set of a given set is the set containing all the possible subsets of the given set.
Let set = S
The power of set is denoted by P(S)

Question 17

Super set def

Answer 17

If every element of A is an element of B then B is set to be super set of A and is written as B _) A.

Question 18

One-one correspondence def

Answer 18

If elements of two sets A and B are paired in such a way that each element of A is paired with one and only one element of B and vice versa, then such a pairing is called one-one correspondence between A and B

Question 19

Order of a set

Answer 19

The number of distinct elements in a finite set A is called the order of set and is denoted by n(A).

Question 20

Universal set def

Answer 20

If all sets under consideration are the subsets of a fixed set U, then U is called univeral set or the universe of discourse

Question 21

Union of sets def

Answer 21

The union of two sets A and B is denoted by A U B and is a set of all elements that belong to A or B.

Question 22

Intersection of two sets def

Answer 22

Intersection of two sets A and B is denoted by A (UPSIDE DOWN U) B and is the set of all elements that belong to A and B.

Question 23

Disjoint sets def

Answer 23

If the intersection of two sets is empty, then sets are said to be disjoined sets.

Question 24

Overlapping sets def

Answer 24

If the intersection of two sets is non-empty but neither is a subset of other thrn sets are said to be overlapping.

Question 25

Complement of a set def

Answer 25

If U is a universal set, then complement of a set A is denoted by A' or A (raise to power c) and is defined as set of all those elements of U that are not in A.

Question 26

Difference of two sets

Answer 26

If A and B are two sets, then the difference of a set A and B is denoted by A-B and is defined as a set of all those elements of A that are not in B.

Question 27

Venn diagram def

Answer 27

A venn diagram is a pectorial representation of sets in which sets are represented by enclosed areas in a plane. They were first used by an English logician and mathematician John Venn

Question 28

Commutative property of union

Answer 28

AUB=BUA

Question 29

Commutative property of intersection

Answer 29

A(UPSIDE DOWN U)B=B(UPSIDE DOWN U)A

Question 30

Associative property of union

Answer 30

AU(BUC)=(AUB)UC

Question 31

Associative property of intersection

Answer 31

A(UPSIDE DOWN U)(B(UPSIDE DOWN U)C)=(A(UPSIDE DOWN U)B)(UPSIDE DOWN U)C

Question 32

Distributivity of union over intersection

Answer 32

AU(B(UPSIDE DOWN U)C)=(AUB)(UPSIDE DOWN U)(AUC)

Question 33

Distributivity of intersection over union

Answer 33

A(UPSIDE DOWN U)(BUC)=(A(UPSIDE DOWN U)B)U(A(UPSIDE DOWN U)C)

Question 34

De Morgan's laws

Answer 34

(AUB)'=A' (UPSIDE DOWN U) B' (A(UPSIDE DOWN U)B)' = A'UB'

Question 35

Logic def

Answer 35

The basic principles and techniques that are used to distinguish correct reasoning from incorrect reasoning.

Question 36

Induction logic def

Answer 36

A process of reasoning that infers ageneral conclusions based on idvidual cases.

Question 37

Deduction logic def

Answer 37

A process of reasoning that starts with general truth and applies that truth to the specific case.

Question 38

Preposition def

Answer 38

A declarative statement that may be true or false but not both is called preposition. 4<5 true 5<4 false

Question 39

Aristotlian logic def

Answer 39

Deductive logic in which every statement is regarded as true or false and there is no other possibility is called aristotlian logic. E.g a =b can either be true or false

Question 40

Non aristotlian logic def

Answer 40

Deductive logic in which for every statement there is a scope for 3rd or 4th possibility is called non aristotlian logic

Question 41

Conjunction def

Answer 41

If p and q are two statements, then a compound statement in form if p then q (p implies q) is called "implication or conditional." Conjuction of two statements p and q is denoted by p ^ q. Conjuction is said to be true if both of its components are true.

Question 42

Disjunction def

Answer 42

The disjunction of two statements is denoted by p \/ q and is said to be true if at least one of its components p and q is true and false when both components are false.

Question 43

Implication or conditional def

Answer 43

A conditional is said to be false only when antecedent is true and consequent is false. In all other cases, it is true. In P-> q p is antecedent, and q is consequent

Question 44

Biconditional (equivalence)

Answer 44

The preposition p->q /\ q-> p is shortly written as p<---> q (p iff q) and is called biconditional or equivalence. P<--> is only true when both p and q are true or both p and q are false.

Question 45

Tautology def

Answer 45

A statement that is true for all positive values of the variables involved in it is called a tautology E.g p\/ ~p is tautology

Question 46

Absurdity def

Answer 46

A statement that is always false is called an absurdity or a contradiction. E.g p/\~p is absurdity

Question 47

Contingency def

Answer 47

A statement that can be true or false depending upon the truth values of the variable involved in it is called a contingency. E.g (p->q)/\ (p\/q)

Question 48

Quantifiers def

Answer 48

The words or symbols that convey the idea of quantity or numbers are called quantifiers.

Question 49

Types of quantifiers

Answer 49

Universal quantifiers Existential quantifiers

Question 50

Universal quantifiers

Answer 50

The symbol \-/ which read "for all" is called the universal quantifier.

Question 51

Existential quantifiers

Answer 51

The symbol -] which read as "there exists" is called existential quantifiers.

Question 52

Cartesian product def

Answer 52

Let A and B be two non emptey sets then cartesian product of A and B is denoted by A×B and is defined as A×B= {(x,y) | x[- A and y [- B }

Question 53

Binary relation

Answer 53

Let A and B be two non empty sets then any subset of cartesian product A×B is called binary relation or simply reslation. It is denoted by " r "

Question 54

Domain and range def

Answer 54

The first element of the ordered pair of a relation is called its domain The second element of the ordered pair of a relation is called its range.

Question 55

Relation on a set def

Answer 55

If A is a non-empty set, then any subset of A×A is called relation in A or relation on A.

Question 56

Function def

Answer 56

Let A and B be two non empty sets such that 1. f is a relation from A to B ghat is, f is a subset of A×B 2. Dom f =A 3. First element of no two pairs of f are equal. Then f is said to be a function from A to B. Function is written as f: A->B

Question 57

Into function def

Answer 57

If a function f:A->B is such that Ran f(is proper subset of )(c) B i.e Ran f ≠ B then f is said to be a function from A into B. A={1,3,5} B={2,4,6,8} f={(1,2),(3,4),(5,6)}

Question 58

Onto (Subjective) function

Answer 58

If a function f:A—> B is such that Rang f= B, i.e., every element of B is the image of some elements of A, then f is called an onto function or subjective function. A={1,2,3} B={4,5} f={(1,4),(2,4),(3,5)}

Question 59

(1-1) and into (injective) function

Answer 59

If a function f from A into B is such that the second elements of no two of its ordered pairs are equal, then it's called injective function. A={1,2} B={a,b,c} f={(1,a), (2,b)}

Question 60

(1-1) and onto (bijective) function

Answer 60

If f is a function from A onto B such that the second elements of nontwo of its ordered pair are the same, then f is said to be bijective function. It is also called (1-1) correspondence between A and B.

Question 61

Linear function

Answer 61

The function {(x,y)|y=mx+c} is called linear function because its graph is a straight line.

Question 62

Quadratic function

Answer 62

The function {(x,y)| y= ax²+bx+c} is calleda quadratic function because it is defined by quadratic equation in x,y.

Question 63

Inverse of a function

Answer 63

The inverse of a function f is denoted by f-¹ and is obtained by interchanging the components of each ordered pair if the function is in tabular form.

Question 64

Identity function

Answer 64

The function {(x,y)|y=x} is called identity function.

Question 65

Square root function

Answer 65

The function {(x,y)|y=/x , x>(or equal to )0}

Question 66

Unary operation

Answer 66

An operation that when performed on a single number yields another number of the same or different system is called unary operation. E.g extraction of square roots or cube roots, squaring a number or raising it to higher power.

Question 67

Binary operation.

Answer 67

Let G be a non-empty, set then binary operation on the set G is a function from set G×G to the set G. For E.g, ordinary addition and multiplication are binary operations on N.

Question 68

Residue class modulo n

Answer 68

Three consecutive numbers may be written in the form 3n, 3n+1, 3n+2 when divided by 3, they give remainders 0,1, and 2, respectively. The set of remainders {0,1,2,} is called residue class modulo 3.

Question 69

. Is there any set which has no proper subset? If so, name that set

Answer 69

Yes. The empty set has no proper subset.