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).

**Example 1:**

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

**Solution:**

(i) **Closure axiom:**

Let *x*, *y* ∈ *G*. Then *x* = *a* + *b* √2, *y* = *c* + *d* √2 ; *a, b, c, d* ∈ *Q*.

*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, y* ∈ *G*. 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 *A* ∈ *G*, exists i.e. *A*^{−1} exists and is of order

2 × 2 and *AA*^{−1} = *A*^{−1}*A* = *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.

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