# Discrete Mathematics | Week 12

## Quiz

1. Which of the following are true?

(Z,×) is a group

(Z,+) is not a group

(Q,×) is a group

(Z,−) is not a group

(R,+) is a group

c.(Q,×) is a group

d. (Z,−) is not a group

e. (R,+) is a group

Explanation

2. Let T be a group of 100 elements, what is the largest possible subgroup of T other than T itself?

a. 50

Explanation

3. The generating function for odd partitions of x is:

Explanation

4. For a path graph of P7, in how many ways can you color this graph with 4 colors so that no two adjacent nodes have the same color?

b. 2916

Explanation

5. What are the elements of Z7, i.e., integer modulo 7?

c. {0, 1, 2, 3, 4, 5, 6}

Explanation

6. Which of the following statement(s) is/are true?
I) The operations of a group and its subgroup can be different.
II) (Q, +) is a subgroup of (R, +)
III) A group under addition modulo n, where n is prime, cannot have a subgroup except for a trivial subgroup identity element alone.

a. II and III

Explanation

7. State whether true/false:
𝑃d(𝑛) is the coefficient of 𝑥n in the function (1+x)(1+x2)(1+x3)……..(1+xn)

a. True

Explanation

8. Let A be a group with 70 elements. Then A cannot have a subgroup of cardinality

c. 20

Explanation

9. (Z6,+) is a group, what is the inverse of 4 in this group?

b. 2

Explanation

10. Given a complete graph with 11 vertices, in how many ways can you color this graph such that no two adjacent vertices have the same color? You are given 26 colors.

c. C(K11)=26×25×24×…×16

Explanation 0 