Problem 66 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Verify the assertion that two sets \(A\) and \(B\) are equal if and only if (1) \(A \subseteq B\) and (2) \(B \subseteq A\).

Step-by-Step Solution

Answer

To verify the assertion that two sets A and B are equal if and only if (1) \(A \subseteq B\) and (2) \(B \subseteq A\), we prove two implications: 1. If A and B are equal, then \(A \subseteq B\) and \(B \subseteq A\). This is true because if \(A = B\), every element of A is an element of B and vice versa. 2. If \(A \subseteq B\) and \(B \subseteq A\), then A and B are equal. This is true since every element in A is an element in B and every element in B is an element in A. Hence, we have proven that sets A and B are equal if and only if (1) \(A \subseteq B\) and (2) \(B \subseteq A\).

Step 1: Prove that if A and B are equal, then \(A \subseteq B\) and \(B \subseteq A\)

Assume that \(A = B\). By definition of set equality, every element of A is also an element of B, and every element of B is also an element of A. This confirms that \(A \subseteq B\) and \(B \subseteq A\).

Step 2: Prove that if \(A \subseteq B\) and \(B \subseteq A\), then A and B are equal

Now, let's assume that \(A \subseteq B\) and \(B \subseteq A\). We need to show that A and B have the same elements, and therefore they are equal sets. First, let \(x\) be an arbitrary element in A. Since \(A \subseteq B\), it must be that \(x\) is also an element of B. Hence, every element in A is also an element in B. Next, let \(y\) be an arbitrary element in B. Since \(B \subseteq A\), it must be that \(y\) is also an element of A. Hence, every element in B is also an element in A. Since every element in A is an element in B and every element in B is an element in A, we can conclude that A and B are equal sets. This proves the second implication.

Step 3: Conclusion

By proving the two implications, we have verified the assertion that two sets A and B are equal if and only if (1) \(A \subseteq B\) and (2) \(B \subseteq A\).

Key Concepts

Set Equality

In set theory, set equality is an important concept that defines when two sets are considered the same. Sets are said to be equal, denoted as \(A = B\), if and only if they have exactly the same elements. This means every element that is in set \(A\) must also be in set \(B\), and vice versa.

When dealing with problems involving set equality, it's crucial to check both directions. If one set has elements that the other does not, the sets are not equal. This mutual containment is also depicted through the concept of subsets, which we will explore in the next section.

Understanding set equality lays the foundation for many mathematical concepts, as it ensures that comparisons between sets are rigorous and precise.

Subset

A subset is a concept where all elements of one set are contained within another set. If every element of set \(A\) is also an element of set \(B\), we express this relationship as \( A \subseteq B \). It's important to note here that the subset can be equal to the set it is compared to. This means, \( A \) could have fewer, or exactly the same, elements as \( B \).

The idea of subsets is crucial for understanding set equality. To prove two sets are equal, you must demonstrate that each is a subset of the other:

• First, show that \(A\) is a subset of \(B\), meaning every element in \(A\) is found in \(B\).
• Then, confirm that \(B\) is a subset of \(A\), ensuring every element in \(B\) is also in \(A\).

By proving both of these, we establish that \(A = B\). Remember, the concept of subsets not only helps in set equality but also forms the backbone of other set operations and relationships.

Mathematical Proof

Mathematical proof is an essential part of confirming the truth of statements in mathematics. It involves a sequence of logical steps that show why something is true. In the context of set theory, proving set equality requires demonstrating two critical subset relations.

Here's how proofs work:

• **Assumption**: Begin by assuming one side of the equation or condition.
• **Logical Argumentation**: Use logical reasoning and known facts or definitions to draw conclusions.
• **Conclusion**: Arrive at a conclusion, demonstrating that the initial assumption leads to the outcome.

To prove that two sets \(A\) and \(B\) are equal, you show that assuming \(A = B\) leads to \(A \subseteq B\) and \(B \subseteq A\), and vice versa. Through clear and logical argumentation, mathematical proofs provide a rigorous way to establish truths in mathematics.

Logic

Logic forms the basis of reasoning in mathematics, allowing us to make valid arguments and conclusions. It is the glue that holds mathematical proofs together. In set theory, logical statements and operators are used to illustrate the relationships between sets.

When asserting that two sets are equal, logic dictates that both sets must contain the same elements. To do this, consider:

• **If-Then Statements**: Used to establish subset conditions like "If \(A\) is a subset of \(B\), then every element of \(A\) is in \(B\)."
• **Equivalences**: Statements that show two conditions are mutually inclusive, such as proving \( A = B \) is equivalent to saying both \( A \subseteq B \) and \( B \subseteq A \).

Being able to follow and construct logical arguments is key to understanding set theory and successfully engaging with problems concerning set equality and other mathematical phenomena.