Discrete Math Proof

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. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. In this article we shall discuss about discrete math proof.(Source: wikipedia)

Discrete math theorem and proof

Theorem 1:

For any group G, the identity element is the only element of order 1.

Proof:

If a (≠ e) is another element of order 1 then by the definition of order of an element, we have (a)1 = e ⇒ a = e which is a contradiction. Therefore e is the only element of order 1.

Theorem 2:

The identity element of a group is unique.

Proof:

Let G be a group. If possible let e1 and e2 be identity elements in G.

Treating e1 as an identity element we have e1 * e2 = e2 …               (1)

Treating e2 as an identity element, we have e1 * e2 = e1 …              (2)

From (1) and (2), e1 = e2

Therefore Identity element of a group is unique.

Theorem 3:

The inverse of each element of a group is unique.

Proof:

Let G be a group and let a ∈ G.

If possible, let a1 and a2 be two inverses of a.

Treating a1 as an inverse of ‘a’ we have a * a1 = a1 * a = e.

Treating a2 as an inverse of ‘a’, we have a * a2 = a2 * a = e

Now a1 = a1 * e = a1 * (a * a2) = (a1 * a) * a2 = e * a2 = a2

⇒ Inverse of an element is unique.

Discrete math proof example problem

Example: Find the order of each element of the group (G, .) where G = {1, − 1, i, − i}.

Solution: In the given group, the identity element is 1. Therefore 0(1) = 1.

0(− 1) = 2 [ Therefore we have to multiply − 1 two times (minimum) to get 1 i.e.,

(− 1) (− 1) = 1]

0(i) = 4 [ Therefore we have to multiply i four times to get 1, i.e., (i) (i) (i) (i) = 1]

0(− i) = 4 [ Therefore we have to multiply − i four times to get 1].

Example 2:

Find the order of each element of the group (G, .) where G = {1, − 1, i, − i}.

Solution: In the given group, the identity element is 1.Therefore 0(1) = 1.

0(− 1) = 2 [Therefore we have to multiply − 1 two times (minimum) to get 1 i.e.,

(− 1) (− 1) = 1]

0(i) = 4 [Therefore we have to multiply i four times to get 1, i.e., (i) (i) (i) (i) = 1]

0(− i) = 4 [Therefore we have to multiply − i four times to get 1].