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Chapter 6: Problem 38

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A \subseteq B\), then \(n(B)=n(A)+n\left(A^{c} \cap B\right)\).

Short Answer

Expert verified
The statement is true. If \(A \subseteq B\), then the number of elements in B equals the sum of the number of elements in A and the number of elements in the intersection of the complement of A and B, or \(n(B) = n(A) + n(A^{c} \cap B)\). This is because B can be considered the union of two disjoint sets A and \(A^{c} \cap B\), for which the cardinality of the union is equal to the sum of their cardinalities.

Step by step solution

01

Understand the terms used in the statement

Try to understand each symbol and term used in the statement: - \(A \subseteq B\) means that A is a subset of B (i.e., all elements of A are also elements of B). - \(n(X)\) indicates the cardinality or the number of elements in the set X. - \(A^{c}\) represents the complement of the set A (i.e., all elements not in A). - \(A^{c} \cap B\) means the intersection of the complement of A and set B (i.e., all elements that belong to both the complement of A and B).
02

Rewrite the statement in a logical order

Rephrase the statement to understand the relation between the terms: "If A is a subset of B, then the number of elements in B equals the sum of the number of elements in A and the number of elements in the intersection of the complement of A and B."
03

Apply set theory concepts

Now, recall the properties of sets and utilize them to prove the statement. Consider B as the union of the two disjoint sets A and \(A^{c} \cap B\), because there are no elements that belong to both A and \(A^{c} \cap B\). \(B = A \cup (A^{c} \cap B)\) For disjoint sets, the cardinality of the union is equal to the sum of their cardinalities: \(n(B) = n(A) + n(A^{c} \cap B)\)
04

Conclude the result

Since we were able to demonstrate that for any A being a subset of B, the cardinality of B is equal to the sum of the cardinalities of A and the intersection of the complement of A with B, we can conclude that the given statement is true.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If a standard minimization linear programming problem has a unique solution, then so does the corresponding maximization problem with objective function \(P=-C\), where \(C=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\) is the objective function for the minimization problem.

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{cccccc|c} x & y & z & u & v & P & \text { Constant } \\ \hline 3 & 0 & 5 & 1 & 1 & 0 & 28 \\ 2 & 1 & 3 & 0 & 1 & 0 & 16 \\ \hline 2 & 0 & 8 & 0 & 3 & 1 & 48 \end{array} $$

Solve each linear programming problem by the method of corners. $$ \begin{array}{l} \text { Maximize } P=4 x+2 y \\ \text { subject to } \quad x+y \leq 8 \\ \quad 2 x+y \leq 10 \\ x \geq 0, y \geq 0 \end{array} $$

A company manufactures products \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Each product is processed in three departments: I, II, and III. The total available labor-hours per week for departments I, II, and III are 900,1080 , and 840 , respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows: $$ \begin{array}{lccc} \hline & \text { Product A } & \text { Product B } & \text { Product C } \\ \hline \text { Dept. I } & 2 & 1 & 2 \\ \hline \text { Dept. II } & 3 & 1 & 2 \\ \hline \text { Dept. III } & 2 & 2 & 1 \\ \hline \text { Profit } & \$ 18 & \$ 12 & \$ 15 \\ \hline \end{array} $$ How many units of each product should the company produce in order to maximize its profit? What is the largest profit the company can realize? Are there any resources left over?

You are given the final simplex tablea for the dual problem. Give the solution to the primal prob lem and the solution to the associated dual problem. Problem: Minimize \(\quad C=2 x+3 y\) subject to \(\begin{aligned} x+4 y & \geq 8 \\ x+y & \geq 5 \\ 2 x+y & \geq 7 \\\ x \geq 0, y & \geq 0 \end{aligned}\) Final tableau: $$ \begin{array}{cccccc|c} u & v & w & x & y & P & \text { Constant } \\ \hline 0 & 1 & \frac{7}{3} & \frac{4}{3} & -\frac{1}{3} & 0 & \frac{5}{3} \\ 1 & 0 & -\frac{1}{3} & -\frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ \hline 0 & 0 & 2 & 4 & 1 & 1 & 11 \end{array} $$
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