Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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 \(n(B)=n(A)+n\left(A^{c} \cap B\right)\).

Step-by-Step Solution

Answer

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 1: 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).

Step 2: 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."

Step 3: 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: \(n(B) = n(A) + n(A^{c} \cap B)\)

Step 4: 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.