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Chapter 2: Problem 3

Recall that the symbol \(\oplus\) denotes the logical exclusive or operation. If \(A\) and \(B\) sets, define the set \(A \triangle B\) by \(A \Delta B=\\{x \mid(x \in A) \oplus(x \in B)\\} .\) Show that \(A \triangle B=(A \backslash B) \cup(B \backslash A) .(A \triangle B\) is known as the symmetric difference of \(A\) and \(B\).)

Short Answer

Expert verified
\(A \triangle B = (A \backslash B) \cup (B \backslash A)\) as both represent elements unique to each set.

Step by step solution

01

Understanding the Symmetric Difference

First, we need to understand that the symmetric difference of two sets \(A\) and \(B\), denoted \(A \triangle B\), consists of all elements that are in either of the sets \(A\) or \(B\) but not in both. In terms of logical operations, this represents elements where either \(x \in A\) is true xor \(x \in B\) is true.
02

Breaking Down the Terms

Next, consider the terms on the right side of the equation: \((A \backslash B)\) and \((B \backslash A)\). \((A \backslash B)\) consists of elements that are in \(A\) but not in \(B\), and \((B \backslash A)\) consists of elements that are in \(B\) but not in \(A\). By combining these terms using a union, \((A \backslash B) \cup (B \backslash A)\) captures all elements unique to each set.
03

Logical Interpretation of Set Difference

For any element \(x\), \(x \in (A \backslash B)\) if \(x \in A\) and \(x otin B\), and \(x \in (B \backslash A)\) if \(x \in B\) and \(x otin A\). Thus, the condition \(x \in (A \backslash B) \cup (B \backslash A)\) is satisfied if \(x\) is in exactly one of the sets \(A\) or \(B\), aligning perfectly with the definition of the symmetric difference.
04

Verification of the Equation

Lastly, to verify \(A \triangle B = (A \backslash B) \cup (B \backslash A)\), let \(x\) be any element. Examine the two scenarios: if \(x otin A\) and \(x \in B\), or \(x \in A\) and \(x otin B\), then \(x\) satisfies exactly one side of the exclusive \(or\), so \(x\) is in \(A \triangle B\). This matches the logic of \(x\) being in exactly one of \(A\) or \(B\), so \(x \in (A \backslash B) \cup (B \backslash A)\). Therefore, \(A \triangle B\) is indeed equal to \((A \backslash B) \cup (B \backslash A)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set Theory
Set Theory is a branch of mathematical logic that studies sets, which are collections of objects. Sets are fundamental objects in mathematics and are used to define all other mathematical structures. In set theory, we explore concepts like unions, intersections, and set differences. These operations help us understand relations and groupings within various elements.
In the context of the symmetric difference, set theory provides us with a way to define new sets based on the relationships between their elements. Here, we have two sets, \( A \) and \( B \), and we're interested in creating a new set, \( A \triangle B \), which consists of elements unique to either \( A \) or \( B \), but not in both. This operation is quite similar to the more commonly known set union, yet it distinguishes itself by excluding the intersection of the two sets.
Understanding these interactions is crucial in various applications, such as solving problems in probability, logic, and even computer science. The beauty of set theory lies in its simplicity and its foundational role in other mathematical disciplines.
Logical Operations
Logical operations are the tools we use to manage and manipulate logical statements. They are essential in determining the truth values of complex statements based on simpler components. In mathematics and computer science, logical operations are vital for constructing conditions and controlling the flow of algorithms.
Among these logical operations, we find 'and', 'or', and 'not'. However, the exclusive or, or XOR, is particularly interesting because it is a bit different from the regular 'or'. Using XOR means that the operation is true only if exactly one of the inputs is true. This distinction makes the exclusive or a powerful concept in logical reasoning.
  • The XOR operation can be represented as \( A \oplus B \).
  • In the symmetric difference, we use this operation to filter out elements that appear in both sets, leaving us with elements exclusive to each set.
Logical operations, especially XOR, allow us to efficiently implement and understand complex logical expressions across various domains. They are the backbone of digital circuits and are used in algorithms for error checking and data processing.
Exclusive Or
Exclusive Or, abbreviated as XOR, is an operation used in various fields, most notably in binary logic and computer science. The term 'exclusive' indicates that the operation yields a true result only when one, and only one, of the operands is true.
When applied to sets, this operation underpins the concept of the symmetric difference. In simple terms, \( A \oplus B \) or \( A \triangle B \) implies elements are in either set \( A \) or set \( B \), but not in both.
  • This makes XOR extremely useful in situations where we only want to combine mutually exclusive conditions.
  • For instance, in digital electronics, XOR gates are critical components for constructing arithmetic logic units, which perform operations like addition.
The XOR operation may seem subtle, but its applications are vast. Understanding XOR allows for deeper insights into how logical evaluations work to control processes, detect errors, and manage data in technology and mathematics.

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Most popular questions from this chapter

Let \(A\) and \(B\) be finite sets with \(|A|=n\) and \(|B|=m\). List the elements of \(B\) as \(B=\left\\{b_{0}, b_{1}, \ldots, b_{m-1}\right\\}\). Define the function \(\mathcal{F}: A^{B} \rightarrow A \times A \times \cdots \times A\), where \(A\) occurs \(m\) times in the cross product, by \(\mathcal{F}(f)=\left(f\left(b_{0}\right), f\left(b_{1}\right), \ldots, f\left(b_{m-1}\right)\right)\). Show that \(\mathcal{F}\) is a one-to-one correspondence.

Consider each of the following relations on the set of people. Is the relation reflexive? Symmetric? Transitive? Is it an equivalence relation? a) \(x\) is related to \(y\) if \(x\) and \(y\) have the same biological parents. b) \(x\) is related to \(y\) if \(x\) and \(y\) have at least one biological parent in common. c) \(x\) is related to \(y\) if \(x\) and \(y\) were born in the same year. d) \(x\) is related to \(y\) if \(x\) is taller than \(y\). e) \(x\) is related to \(y\) if \(x\) and \(y\) have both visited Honolulu.

In the English sentence, "She likes men who are tall, dark, and handsome," does she like an intersection or a union of sets of men? How about in the sentence, "She likes men who are tall, men who are dark, and men who are handsome?"

Let \(A\) be the set \(\\{a, b, c, d\\}\). Let \(f\) be the function from \(A\) to \(A\) given by the set of ordered pairs \(\\{(a, b),(b, b),(c, a),(d, c)\\}\), and let \(g\) be the function given by the set of ordered pairs \(\\{(a, b),(b, c),(c, d),(d, d)\\} .\) Find the set of ordered pairs for the composition \(g \circ f\).

Use induction to prove the following generalized DeMorgan's Law for set theory: For any natural number \(n \geq 2\) and for any sets \(X_{1}, X_{2}, \ldots, X_{n}\), $$ \overline{X_{1} \cap X_{2} \cap \cdots \cap X_{n}}=\overline{X_{1}} \cup \overline{X_{2}} \cup \cdots \cup \overline{X_{n}} $$
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