Nature of Mathematics - 12th Edition

Problem 1-2

Problem 1 can be phrased in your own words, but it would be a good idea to remember De Morgan's laws.

Problem 2 does not have a right or wrong answer, but it should outline the the steps shown in Example 3.

Problems 3-10

Draw a rectangle to represent the universe along with three interlocking sets, as shown in Example 3. Next, take care about the order of operations:

Problems 3 and 4: parentheses first
Problems 5 and 6: complement first
Problems 7 and 8: parentheses first, then complement
Problem 9: parentheses first; then intersection; finally complement
Problem 10: parentheses first; that is, complement of B, then complement of C; next intersection. Finish up with the complement of A, and finally the union.

Problems 11-18

These problems are designed to help you with terminology. Look at the terminology illustrated in Example 1.

Problems 19-22

See Example 1; watch the order of operations:

Problem 19: complement of A first; then B; finally union.
Problem 20: A first; then complement of B; finally intersection.
Problem 21: A intersect B first; then complement.
Problem 22: A function B first; then complement.

Problems 23-34

See Example 3; watch the order of operations

Problems 35-41

Problems 35-38; draw a rectangle for the universe, and then draw overlapping circles for the given sets. Label the universe and each circle.
Problem 39; see Example 5, but in addition to labeling the sets, you must also calculate the percentages. For example 80% of 263 million is
0.80(263) = 210.4 million
Problem 40; Draw three interlocking circles, and let one be vote yea for Bill A, another be vote yea for Bill B, and the third be vote yea for Bill C. (See Figure 2.14)
Problem 41; Draw three interlocking circles, as shown in Figure 2.14.

Problems 42-45

Study each Venn diagram; how many sets? What is the final region?
Hypothesize the relationship, and then check by carrying out the operations as shown in Example 3.

Problems 46-51

Draw a Venn diagram for the left side of the equality. Then, draw a Venn diagram for the right side of the equality. If the final shaded answers for the two Venn diagrams are the same, then you have proved the result. If they are not the same, then you have disproved the result. See Examples 2 and 4.

Problems 52-55

These are survey problems with three sets, like Example 5. Draw three overlapping circles and label each circle.
Step 1: fill in the number of elements in the innermost set.
Step 2: fill in numbers in the other overlappling sections.
Step 3: fill in the other regions.
Step 4: Fill in the number outside the three circles and answer the questions asked in the problem.