Problem 72 - 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 \(A \cap B=A\).

Step-by-Step Solution

Answer

The statement is true. If a set \(A\) is a subset of set \(B\) (\(A \subseteq B\)), then the intersection of \(A\) and \(B\) (\(A \cap B\)) will always be equal to set \(A\). This is because all the elements in set \(A\) are also present in set \(B\) and their intersection will consist of all the common elements, which are the elements of set \(A\).

Step 1: Understand the statement

First, let's understand the given statement. It says that if \(A\) is a subset of \(B\) (denoted as \(A \subseteq B\)), then the intersection of \(A\) and \(B\) (denoted as \(A \cap B\)) is equal to \(A\).

Step 2: Test the statement with an example

Let's test the statement with an example to see if it is true or false. Take two sets A and B such that \(A = \{1, 2\}\) and \(B = \{1, 2, 3, 4\}\). In this case, \(A \subseteq B\) because every element in set \(A\) is also present in set \(B\). Now let's find the intersection of \(A\) and \(B\). The intersection of two sets consists of all the elements that are common to both sets. \(A \cap B = \{1, 2\}\) Comparing the intersection with set \(A\), we find that \(A \cap B = A\), which supports the statement.

Step 3: Provide a logical explanation

Let's provide a logical explanation for the statement. Since \(A\) is a subset of \(B\), it means that all the elements present in set \(A\) are also present in set \(B\). When we find the intersection of the two sets, we are finding the common elements between them. Since all elements of the \(A\) are also in the \(B\), they are the common elements between \(A\) and \(B\). Therefore, the intersection of \(A\) and \(B\) will always be equal to \(A\).

Step 4: Conclude the result

Based on the example and the logical explanation, we can conclude that the given statement is true. If \(A \subseteq B\), then \(A \cap B=A\).

Key Concepts

Subset

In set theory, a subset is a fundamental concept. When we say that set \(A\) is a subset of set \(B\) (written as \(A \subseteq B\)), we mean that every element in set \(A\) is also found in set \(B\). - This does not mean \(A\) is identical to \(B\); \(B\) could have additional elements. - For example, if \(A = \{1, 2\}\) and \(B = \{1, 2, 3, 4\}\), \(A\) is a subset of \(B\) because elements 1 and 2 are in both sets, and \(B\) has more elements than \(A\).Understanding subsets is important as it helps establish logical relationships between different sets in mathematics.

Intersection of Sets

The intersection of sets is a method to find common elements shared by those sets. The symbol for intersection is \(\cap\). For sets \(A\) and \(B\), their intersection, denoted as \(A \cap B\), will include all elements that are in both \(A\) and \(B\). - If \(A = \{1, 2\}\) and \(B = \{1, 2, 3, 4\}\), then the intersection \(A \cap B = \{1, 2\}\). - This set includes only elements that \(A\) and \(B\) both contain.Understanding intersection is crucial for solving problems involving commonalities across sets, and it is especially relevant in proofs concerning subsets and set equality.

Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth or falsehood of a given statement based on established assumptions and theorems. In this context, we used proving techniques to demonstrate that if \(A \subseteq B\), then indeed \(A \cap B = A\). - The proof involves understanding that all elements of \(A\) are in \(B\), thus making any intersection simply yield \(A\). - Moreover, since there are no elements in \(A\) that could be missing in \(B\), the intersection doesn't exclude any elements of \(A\), maintaining the equality.These proofs are often fundamental to ensure the statement holds universally, regardless of any specific numerical example provided.

Logical Explanation

Logical explanations in mathematics use reasoning to make intuitive conclusions understandable. For the statement \(A \subseteq B\) implies \(A \cap B = A\), we use logical reasoning to see why this holds true. - Since \(A\) is a subset of \(B\), every element in \(A\) must appear in \(B\). - Therefore, when we contemplate the intersection \(A \cap B\), all elements of \(A\) appear in the result, making the intersection equivalent to \(A\).Logical explanations bridge the gap between abstract mathematical concepts and a student's understanding, ensuring that conclusions are comprehensible and valid.