5. Define the following sets: A = {1, 2, 3} B = {1, 2} Recall that elements of t

Question

5. Define the following sets:

            A = {1, 2, 3}

            B = {1, 2}

Recall that elements of the cross product of two sets are ordered pairs.

a. Give one element of A x B that is not in B x A, or explain why no such element exists.

b. Give one element of B x A that are not in A x B, or explain why no such element exists.

c. How many elements are there in A x B?

d. How many elements are there in B x A?

e. Explain why A x A is or is not a subset of A x B.

6. Use the ceiling or floor function to give a mathematical expression:

There are r runners on the cross country team. Each runner needs three bottles of water for a meet. A crate has 24 bottles. How many crates must be purchased for the meet? State your answer as a function of r.   

7. Define the following functions f, g, and h:

•f: {0, 1}3{0, 1}3. The output of f is obtained by taking the input string and replacing the first bit by 1, regardless of whether the first bit is a 0 or 1. For example, f(001) = 101 and f(110) = 110.

•g: {0, 1}3{0, 1}3. The output of g is obtained by taking the input string and reversing the bits. For example, g(011) = 110.

•h: {0, 1}3{0, 1}3. The output of h is obtained by taking the input string x, and replacing the last bit with a copy of the first bit. For example, h(011) = 010.

What is g f(010)?

(b) What is g h(010)?

(c) What is h f(010)?

(d) What is the range of h f?

(e) What is the range of g f?

Explanation / Answer

(According to Chegg policy only one question with four subquestions will be answered. Pleast post the remaining in another question)

            A = {1, 2, 3}

            B = {1, 2}

a. Give one element of A x B that is not in B x A, or explain why no such element exists.

We have A x B = {(1,1),(1,2),(2,1),(2,2),(3,1),(3,2)} and B x A = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)}

An example is (3,1) which is in A x B but not in B x A.

b. Give one element of B x A that are not in A x B, or explain why no such element exists.

An example is (1,3) which is in B x A but not in A x B.

c. How many elements are there in A x B?

There are 6 elements in A x B

d. How many elements are there in B x A?

There are 6 elements in B x A

e. Explain why A x A is or is not a subset of A x B.

Since A has 3 elements, A x A will have 9 elements. On the other hand, A x B has only 6 elements. So A x A cannot be a subset of A x B.

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