Section 5.1 Homework Hints

Problems 1-9

There are many problems throughout the text labeled IN YOUR OWN WORDS. Just relax; do not be afraid to give your opinion. For the most part, these questions do not have "right" or "wrong" answers. You should spend about five minutes in answering each of these questions.

Problems 10-15

Addition is accepted as an undefined term, but multiplication is defined as repeated addition.  You need to carefully apply this definition for these problems.  For example, the answer to Problem 10a is not 6, but rather is 2*3 = 3 + 3.

Problems 16-26

Review the definition of the associative and commutative properties. Remember that when the commutative property is used, the order in which the elements appear from left to right is changed, but the grouping is not changed. When the associative property is used, the elements are grouped differently, but the order in which they appear is not changed.

Problems 27-30

These problems are designed to test your understanding the commutative and associative properties in interesting everyday settings.

Problem 31

There are many problems throughout the text labeled IN YOUR OWN WORDS. Just relax; do not be afraid to give your opinion. For the most part, these questions do not have "right" or "wrong" answers. You should spend about five minutes in answering each of these questions.

Problems 32-33

These problems are discussing operations defined by table. Remember to read the column at the left first.  In Problem 33, (-1) x i is located in row 2 column 3.  On the other hand i x (-1) is located in row 3 column 2.

Problems 34-35

See Examples 1 and 2. A set is not closed for an operation if you can find one example of an answer that is not in the set. To show that a set is closed for an opertaion, you must look at the table of all outcomes.

Problems 36-41

These problems are defining a variety of operations. Make sure you understand each defined operation before attempting to answer the question in the problem.

Problems 42-43

a(b + c) = ab + ab is the distributive property for multiplication over addition.
We apply this property to the given operations:

Problem 42: right over down would be:
a right (b down c) = (a right b) down (a right c)
Apply this for at least three sets of three natural numbers.

Problem 43: down over right would be:
a down (b right c) = (a down b) right (a down c)
Apply this for at least three sets of three natural numbers.

Problems 44-45

Use the distributive property (see Example 4) for mental operations as in:

             7 * 82 = 7*(80 + 2) = 7*80 + 7*2 = 560 + 14 = 574

You can do the above steps mentally.

Problem 46

Construct a multiplication table for the given set, and then look to see if the reversal of entries gives the same result.

Problem 47

You need to try all possible ordered triplets (there are 27 possibilities). If you try several and look for patterns, you should be able to form a conclusion without trying all of them. However, to be absoutely certain, you will need to try all possibilities.

Problem 48

Construct a multiplication table for the given set, and then look to see if the reversal of entries gives the same result.

Problems 49-52

Remember, the even numbers are numbers of the form 2n; namely,
{2, 4, 6, ... }.  The odd numbers are of the form 2n + 1; namely, {1, 3, 5, ... }.