Author Archives: matchboxcreations777

GROUPS

Definition of Groups

A group G is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.

G1(closure): for all X and Y in G

X*Y is in G

G2(Associativity):For all X,Y and Z in G

(X*Y)*z=X*(Y*Z).

G3(Identity). There is an element e in G such that

e*X=X*e=X

for all X in G.

G4(inverse).for all X in G there is an X’ in G such that

X*X’=X’*X=e

Examples of Groups

1.The integers modulo n, often denoted by Zn, form a group under addition.

2.Group of integers Z with addition as operation and zero as identity element.

3 . The rational numbers form a group under addition.

4 . The positive rationals form a group under multiplication with identity element 1.

5 . The positive rationals form a group under multiplication with identity element 1

6 . The real numbers form a group under addition.

7 . The complex numbers form a group under addition.

8 . The set of all non-zero real numbers is group under the operation of multiplication.

9 . The set of all non-zero complex numbers is group under the operation of multiplication.

10. The set of all 2×2 matrices

[a b]

[c d]

with real (or complex) entries satisfying ad−bc 6= 0 is a group under the operation of matrix multiplication

11 . The set of all 2×2 matrices

[a b]

[c d]

with real (or complex) entries satisfying ad−bc = 1 is a group under the operation of matrix multiplication.

12 . The set of all non-zero rational numbers is group under the operation of multiplication

PROBLEM 3

EXERCISE 3.1

Construct the truth table for the statement(¬p)^q.

     P       q    ¬p (¬p)^q
     T       T      F      F
     T       F      F      F
     F       T      T      T
     F       F      T       F

EXERCISE 3.2

By constructing the truth table, show that ¬(¬p) is logically equivalent to p.∨

     P       ¬p    ¬(¬p)
     T       F      T
     F       T      F

EXERCISE 3.3

Construct the truth tables for p⇒(q∨r).Is the statement logically equivalent to (p⇒q)∨(p⇒r)

     p q r q∨r P⇒( q∨r) p ⇒q p⇒r (p ⇒q)∨(p⇒r)
T T T T T T T T
T T F T T T F T
T F T T T F T T
F T T T T T T T
F F T T T T T T
F T F T T T T T
T F F F F F F F
F F F F T T T T

EXERCISE 3.4

Write down the converse of the following statement

Q.   If n is the multiple of 3 then n is not the multiple of 7

Ans: If n is not the multiple of 7 then n is the multiple of 3.

EXERCISE 3.5

Write down the contrapositive forms of the statements

Q.     If n is a multiple of 7 then n is not a multiple of 3

Ans: If n is a multiple of 3 then n is not a multiple of 7

Q.   If n is a multiple of 12 then n is a multiple of 4

Ans:  If n is not a multiple of 4 then n is not multiple of 12.

PROBLEM 2

PROBLEM 2.1

1.Write down the mathematical notation for:

(a) the set of all even numbers

X=x|x=2y ∀ y∈Z

(b) the set of all natural numbers whose square is less than 21

X=x|x^2 is greater than 21

2.Give a definition of set in Problem.1(b) by means of a list

Set is the collection of well-defined objects.All the objects containing in a set is called as elements. we can represent a set as

{1,2,3,4…..}

3.In each of the following examples say whether the statement x ∈ X is true or false.

(a) x=273,   X={n ∈ N|n is an odd number}

Answer is TRUE

Because an odd number is defined as, 2 multiplied by any number and add one to it which means  (2n+1).so in the case of 273 it can be expressed as 2*(136)+1=273.

(b) x= 273,  X={n ∈ N|n is a prime number}

Answer is FALSE

Because a prime number is defined as, a number which is divided by one and itself only so when we consider 273 ,it can be divisible by 3 which means that the number is not a prime number

(c) x=112279,  X={n ∈ N|n is a perfect square}

Answer is FALSE

Because a perfect square is defined as an integer which is represented by another number with that many number of times .so we can say that 112279 is not a perfect square.

PROBLEM 2.2

1. There are eight different subsets of the set {a,b,c}.Make a list of them.

{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c},null.

PROBLEM 2.3

1.Construct a counter-example for the following statement

for any set A,B,C  A∩(B∪C) = (A∩B)∪C

This is known as distributive property

here  we take A,B,C as each distinct elements which are different.we can solve this property using truth table.