Discrete Mathematics Test 3

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Discrete Mathematics Test 3

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following is/are true about the partial ordering relation?

A
It is reflexive

B
It is asymmetric

C
It is anti-symmetric

D
It is transitive
Solution
Partial ordering relation is
- Reflexive
- Anti-symmetric
- Transitive
It is not asymmetric.
Question 2
1 / -0

Let H be a group with 33 elements and let S be a subgroup of H also H ≠ S. If the size of S is greater than 3 then what is the size of S?

A
11-11

B
114-111

C
117-116

D
119-112

Solution

Lagrange’s theorem: Order of subgroup divides order of group.
Divisor of 33: 1, 3, 11 and 33
∴ size of S must be 1, 3, 11 or 33
Since in the question it said that size is greater than 3 and not equal to 33
Hence size of S is 11

Question 3

1 / -0

If {(0, 1, 2, 3, 4, 5), ⊕6} is a group then which of the following is/are true about the inverse of group's element where ⊕6 denotes addition modulo 6?

Inverse of 2 is 3

Inverse of 4 is 2

Inverse of 3 is 3

Inverse of 5 is 4

Solution

⊕6	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	0
2	2	3	4	5	0	1
3	3	4	5	0	1	2
4	4	5	0	1	2	3
5	5	0	1	2	3	4

Inverse of 0 is 0 from table

Or By

(Inverse+ 0) % 6= 0 ∴ inverse = 0

Similarly

Inverse of 1 is 5

Inverse of 2 is 4

Inverse of 3 is 3

Inverse of 4 is 2

Inverse of 5 is 1

Tips:

{(0, 1, 2, 3, 4, … n – 1, ⊕n} is a group

Question 4
1 / -0

What is the number of generators in the group ({1, ω, ω²}, *) where ω and ω² are cube root of unity and * is multiplication?

A
2-2

B
24-21

C
27-26

D
29-22

Solution

Identity (e) = 1
From 1 we cannot generate ω and ω²
∴ it is not a generator
(ω)¹ = ω, (ω)² = ω² and (ω)³= 1
∴ it is a generator
(ω²)¹ = ω², (ω²)² = ω and (ω²)³ = 1
∴ it is generator
There are two generators
Tips and Tricks
({1, ω, ω²}, *) is a cyclic group
∴ number of generator = ϕ (3) = 3 - 1 = 2
Question 5
1 / -0

Which of the following statement is/are correct?

A
The intersection of any two subgroups of a group G is also subgroup of G.

B
The union of any two subgroups of a group G is also a subgroup of G.

C
The union of two subgroups H1 and H2 where H1 ⊆ H2 of a group (G, * ) is also subgroup of G.

D
Every subgroup of an abelian group is also an abelian group.

Solution

1. The intersection of any two subgroups of a group G is also subgroup of G.
From above example, it is correct. As H1 ꓵ H2 = {1}.
2. The union of any two subgroups of a group G is also a subgroup of G.
Consider group G = {1, 3, 5 ,7 } w.r.t multiplicative(8)
H1 = {1, 3}
H2 = {1, 5}
H1 U H2 = {1, 3, 5}
This is not a subgroup of G. As (3 * 5) mod 8 = 7 which is not in H1 U H2.
3. The union of two subgroups H1 and H2 of a group (G, * ) is also subgroup of G.
This statement is correct when either H1 is a subset of H2 or H2 is a subset of H1.
4. Every subgroup of an abelian group is also an abelian group
Real numbers are abelian group under addition . Every subgroup of abelian group is normal so, each subgroup rise to a quotient group and subgroups, quotients of abelian group are abelian. So, given statement is correct.