Mini Test #2 (Solutions)

1. Which of the following are operations over (the set of positive rational numbers)?

(a)

(b)

(c)

Solution. (a) and (b) are not operations, while (c) is.

2. Suppose , and suppose that for and we know that is a bijection, , and . Find .

Solution. Notice that since is a bijection, it has to be that . Consequently, and so .

3. Suppose and suppose the operation is given by the following Cayley table.

(a) Is the operation associative?

(b) Is the operation commutative?

(c) Is there an identity element?

(d) Which elements are invertible?

Solution. (a)=yes; (b)=no; (c) =no and (d)=none.

4. Find the image of 2 under the following product (composition) of cycles: (2 3 4)(1 3 2)(2 4).

Solution. .