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PROGRAM TITLE: HIGHER NATIONAL DIPLOMA IN COMPUTING
UNIT TITLE: UNIT 18 – DISCRETE MATHS
ASSIGNMENT NUMBER: 04
ASSIGNMENT NAME: ABSTRACT ALGEBRA
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PART 1
When considering the set of all the natural numbers (N), show whether the mathematical
operations of addition, subtraction, multiplication, and division are:
a. Associative
We have the following property: ∀ a,b,c ∈ X , we have a * (b * c ) = ( a * b ) * c
Examples with each mathematical operation:
o Addition : 1 + 2 + 3 = (1 + 2) + 3 = 1 + (2 + 3)
o Subtraction : 3 – 2 – 1 = (3 – 2) – 1 ≠ 3 – (2 – 1)
o Multiplication : 1 × 2 × 3 = (1 × 2) × 3 = 1 × (2 × 3)
o Division : 3 ÷ 2 ÷ 1 = (3 ÷ 2) ÷ 1 ≠ 3 ÷ (2 ÷ 1)
Addition and multiplication are associative operators, and subtraction and division are not
fully associative.
b. Commutative
We have the following property: ∀𝑎, 𝑏 ∈ 𝑋, we have a * b = b * a
Examples with each mathematical operation:
o Addition : 1 + 2 = 2 + 1
o Subtraction : 1 – 2 ≠ 2 – 1
o Multiplication : 1 × 2 = 2 × 1
o Division : 1 ÷ 2 ≠ 2 ÷ 1
In conclusion, addition and multiplication are commutative operators. While subtraction
and division are not.
PART 2
1. Build up the operation tables for group G with orders 1, 2, 3, and 4 using the elements
a, b, c, and e as the identity element in an appropriate way
We show by * an operation of the set G. We show that the set cannot have elements of claim
- That is, if b^3 = e, the operation described is wrong. We conclude that the total number of
potential meanings of the * operation on the set G is less than 9.
e e a b c
a a e c b
b b a c e
c c b e a
The last option table is for collecting claim 4. Assuming we consider only the first row and the
first shard, we get a table to collect claim 1 and assume that we keep the same two-line and
initial section, we get a table to collect claim 4 (see Lagrange's hypothesis). So we present a
table to collect the third request independently:
∗ e a b
e e a b
a a b b
b b e a
a) State the Lagrange's theorem of group theory
To show that (𝐺,∗) is an Abelian group, we will have to prove the following properties:
o Associative :
o Identity element : We will prove that 0 is the identity for S :
∀𝑎 ∈ 𝑆, we have 0 ∗ 𝑎 = 0 + 𝑎 + 0 · 𝑎 = 𝑎 = 𝑎 + 0 + 𝑎 · 0 = 𝑎 ∗ 0.
o Inverse element :
−𝑎
1 +𝑎
∈ ℝ since 𝑎 ≠ − 1. Moreover, if
−𝑎
1 +𝑎
= − 1 , then −𝑎 = − 1 − 𝑎, which is impossible. Thus
−𝑎
1 +𝑎
Then we can compute:
2
o Commutative:
Thus making it commutative in ℝ.
Combining the four proofs above, we can conclude that (𝐺,∗) is an Abelian group.
PART 4
I have prepared a presentation about:
An explanation that adequately explains why group theory is taught to computing students.
The application of group theory within public-key cryptography.
The presentation will take about 15 minutes and will be presented by me.
