Graduate Study In Mathematics

Introduction

               Graduate study in mathematics includes differential calculus, integral calculus, differential equation, and discrete mathematics etc., I this article we shall study about graduate discrete mathematics problems. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics. Such as integers, graphs, and statements in logic.(Source: wikipedia).

 

Graduate study in discrete mathematics example

 

Example 1:

Show that the set G = {a + b √2 / a, bQ} is an infinite abelian group with respect to addition.

Solution:

(i) Closure axiom:

Let x, yG. Then x = a + b √2, y = c + d √2 ; a, b, c, dQ.

x + y = (a + b √2) + (c + d √2) = (a + c) + (b + d) √2 ∈ G, since (a + c) and (b + d) are rational numbers.

Therefore G is closed with respect to addition.

(ii) Associative axiom:

Since the elements of G are all real numbers, addition is associative.

(iii) Identity axiom:

There exists 0 = 0 + 0√ 2 ∈ G such that for all x = a + b √2 ∈ G,

x + 0 = (a + b √2) + (0 + 0√ 2)

= a + b √2 = x

Similarly, we have 0 + x = x. ∴ 0 is the identity element of G and satisfies the identity axiom.

(iv) Inverse axiom:

For each x = a + b √2 ∈ G, there exists − x = (− a) + (− b) √2 ∈ G such that x + (− x) = (a + b √2) + ((− a) + (− b) √2)

= (a + (− a)) + (b + (− b)) √2 = 0

Similarly we have (− x) + x = 0

Therefore (− a) + (− b) √2 is the inverse of a + b √2 and satisfies the inverse axiom.

Therefore G is a group under addition.

(v) Commutative axiom:

x + y = (a + c) + (b + d) √2 = (c + a) + (d + b) √2

= (c + d √2) + (a + b √2)

= y + x, for all x, yG. Therefore the commutative property is true.

Therefore (G, +) is an abelian group. Since G is infinite, we see that (G, +) is an infinite abelian group.

Example 2:

Show that the set of all 2 × 2 non-singular matrices forms, prove the Inverse axiom

Inverse axiom:

 the inverse of AG, exists i.e. A−1 exists and is of order

2 × 2 and AA−1 = A−1A = I. Thus the inverse axiom is satisfied. Hence the set of all 2 × 2 non-singular matrices forms a group under matrix multiplication. Further, matrix multiplication is non-commutative (in general) and the set contain infinitely many elements.

 

Graduate study in discrete mathematics practice problem

 

Let G be the set of all rational numbers except 1 and * be defined on G by a * b = a + bab for all a, bG. Show that (G, *) is an infinite abelian group.