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Discrete Mathematics and its Applications

Kenneth H. Rosen

$Math Studyset Vaia Explanations$ Math

7th Edition · 808 pages

ISBN: 9780073383095

Chapter 2: Q9SE (page 187)

Suppose that A, B, and C are sets. Prove or disprove that $(A - B) - (C - D) = (A - C) - (B - D)$

Short Answer

Expert verified

$(A - B) - (C - D) = (A - C) - (B - D)$

Step by step solution

Step: 1

No, For example, let $A = B = \{a, b\}$ , $C = \emptyset$ and role="math" localid="1668428238907" $D = \{a\}$ . Then $(A - B) - (C - D) = \emptyset - \emptyset = \emptyset$ butrole="math" localid="1668428272716" $(A - C) - (B - D) = \{a, b\} - \{b\} = \{a\}$

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Most popular questions from this chapter

a) Define what it means for a function from the set of positive integers

to the set of positive integers to be one-to-one

b) Define what it means for a function from the set of positive integers to the set

of positive integers to be onto.

c) Give an example of a function from the set of positive integers to the set of

positive integers that is both one-to-one and onto.

d) Give an example of a function from the set of positive integers to the set of

positive integers that is one-to-one but not onto.

e) Give an example of a function from the set of positive integers to the set of

positive integers that is not one-to-one but is onto.

f) Give an example of a function from the set of positive integers to the set of

positive integers that is neither one-to-one nor onto.

Use the identity $\frac{1}{(k (k + 1))} = \frac{\frac{1}{k - 1}}{(k + 1)}$ and Exercise 35 to compute $\sum_{k = 1}^{n} \frac{1}{(k (k + 1))}$

Let $f : R \to R$ and let $f (x) > 0$ for all $x \in R$ . Show that f(x) is strictly decreasing if and only if the functionrole="math" localid="1668414965156" $g (x) = 1 / f (x)$ is strictly increasing.

Which function in exercise 12 is onto?

Find the value of each of these sums.

$\begin{aligned} (a) \sum_{j = 0}^{8} (1 + (- 1)^{j}) \\ (b) \sum_{j = 0}^{8} (3^{j} - 2^{j}) \\ (c) \sum_{j = 0}^{8} (2 \cdot 3^{j} + 3 \cdot 2^{j}) \\ (d) \sum_{j = 0}^{8} (2^{j + 1} - 2^{j}) \end{aligned}$

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Suppose that A, B, and C are sets. Prove or disprove that $(A - B) - (C - D) = (A - C) - (B - D)$

Short Answer

Step by step solution

Step: 1

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Suppose that A, B, and C are sets. Prove or disprove that(A-B)-(C-D)=(A-C)-(B-D)

Short Answer

Step by step solution

Step: 1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Statistics

Probability and Statistics

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Logic and Functions

Study anywhere. Anytime. Across all devices.

Suppose that A, B, and C are sets. Prove or disprove that $(A - B) - (C - D) = (A - C) - (B - D)$