Problem 3
Question
If \(A\) and \(B\) are any two sets, their symmetric difference is the set \(A+B\) defined as follows: $$ A+B=(A-B) \cup(B-A) $$ The shaded area is \(A+B\) It is perfectly clear that \(A+B=B+A\); hence this operation is commutative. It is also associative, as the accompanying pictorial representation suggests: Let the union of \(A, B\), and \(C\) be divided into seven regions as illustrated. \(A+B\) consists of the regions \(1,4,3\), and 6 \(B+C\) consists of the regions \(2,3,4\), and 7 \(A+(B+C)\) consists of the regions \(1,3,5\), and \(7 .\) \((A+B)+C\) consists of the regions \(1,3,5\), and 7 Thus, \(A+(B+C)=(A+B)+C\). If \(D\) is a set, then the power set of \(D\) is the set \(P_{D}\) of all the subsets of \(D\). That is, $$ P_{D}=\\{A: A \subseteq D\\} $$ The operation \(+\) is to be regarded as an operation on \(P_{D}\). Prove the following : Let \(D\) be the three-element set \(D=\\{a, b, c\\} .\) List the elements of \(P_{D}\). (For example, one element if \(\\{a\\}\), another is \(\\{a, b\\}\), and so on. Do not forget the empty set and the whole set \(D\).) Then write the operation table for \(\left\langle P_{D},+\right\rangle\)
Step-by-Step Solution
Verified Answer
The power set is: \( \emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\} \). The table verifies the commutative and associative properties of symmetric difference.
1Step 1: Finding the Power Set
For a set with three elements \(D = \{a, b, c\}\), the power set \( P_D \) includes all possible subsets of \(D\). Since \(D\) has three elements, there are \(2^3 = 8\) subsets. These subsets are: \( \emptyset \), \( \{a\} \), \( \{b\} \), \( \{c\} \), \( \{a, b\} \), \( \{a, c\} \), \( \{b, c\} \), and \( \{a, b, c\} \).
2Step 2: Understanding Symmetric Difference
The operation \( + \) represents the symmetric difference, meaning for two sets \( A \) and \( B \), \( A + B = (A - B) \cup (B - A) \). This operation involves elements in either set but not in both.
3Step 3: Construct Operation Table for Symmetric Difference
We need to construct a table where each cell \( (A, B) \) shows the result of \( A + B \) for all subsets \( A, B \in P_D \). Calculate each entry using the symmetric difference definition for all subset pairs from \(P_D\).
4Step 4: Calculate Table Entries
Here we'll compute the symmetric differences:- \(\emptyset + \emptyset = \emptyset\)- \(\emptyset + \{a\} = \{a\}\), \(\emptyset + \{b\} = \{b\}\), \(\emptyset + \{c\} = \{c\}\),- \(\{a\} + \{a\} = \emptyset\), \(\{a\} + \{b\} = \{a, b\}\),- Continue this pattern for all pairs.Fill the table complete through pairing each subset in \(P_D\) against each other.
5Step 5: Verify Commutativity and Associativity
Verify the operation table exhibits commutative property (\(A + B = B + A\) for all subsets) and associative property (\((A + B) + C = A + (B + C)\)). For these properties, swapping order or grouping doesn't change the result.
Key Concepts
Set TheoryPower SetAssociative PropertyCommutative Property
Set Theory
Set theory is the mathematical study of collections of objects, which we call sets. It provides a foundation for most of modern mathematics by focusing on the basic concept of a set and the relationships between sets. Set theory defines various operations to work with sets including union, intersection, and difference.
In the context of symmetric difference, which is the focus here, we have the operation that combines elements from two sets but excludes those that are common in both. This is denoted as \( A + B = (A - B) \cup (B - A) \).
Understanding set theory is crucial because it lays the groundwork for many mathematical concepts, allowing us to manipulate and understand groups of objects as a single entity. These operations help in solving complex problems by breaking them into manageable parts.When dealing with symmetric differences, we effectively calculate the union of two set differences, providing a result that highlights unique elements between sets.
In the context of symmetric difference, which is the focus here, we have the operation that combines elements from two sets but excludes those that are common in both. This is denoted as \( A + B = (A - B) \cup (B - A) \).
Understanding set theory is crucial because it lays the groundwork for many mathematical concepts, allowing us to manipulate and understand groups of objects as a single entity. These operations help in solving complex problems by breaking them into manageable parts.When dealing with symmetric differences, we effectively calculate the union of two set differences, providing a result that highlights unique elements between sets.
Power Set
A power set is a concept in set theory defined as the set of all subsets of a given set. If you have a set \( D \) that contains elements \( \{a, b, c\} \), the power set \( P_D \) is all possible subsets you can form with these elements.
For a set \( D \) with three elements, the power set includes \( 2^3 = 8 \) subsets, which are:
It forms a comprehensive structure allowing the application of operations like symmetric difference across every possible subset. This is particularly useful when calculating with relational or combinatorial methods in mathematics.
For a set \( D \) with three elements, the power set includes \( 2^3 = 8 \) subsets, which are:
- \(\emptyset\)
- \(\{a\}\)
- \(\{b\}\)
- \(\{c\}\)
- \(\{a, b\}\)
- \(\{a, c\}\)
- \(\{b, c\}\)
- \(\{a, b, c\}\)
It forms a comprehensive structure allowing the application of operations like symmetric difference across every possible subset. This is particularly useful when calculating with relational or combinatorial methods in mathematics.
Associative Property
The associative property refers to the idea that the grouping of three or more elements in an operation does not affect the result. For example, for the symmetric difference operation, \((A + B) + C = A + (B + C)\).
This property is crucial in ensuring that the outcome remains constant regardless of how we associate or group subsets when performing the operation.
When we consider the symmetric difference in terms of the regions mentioned, both groups of calculations \( (A + B) + C \) and \( A + (B + C) \) include regions \(1, 3, 5,\) and \(7\). This consistency demonstrates how the associative property guarantees reliability, even when the set operations become complex.
Math operations like addition and multiplication also exhibit this property, ensuring predictable and stable results no matter how terms are grouped within the operation.
This property is crucial in ensuring that the outcome remains constant regardless of how we associate or group subsets when performing the operation.
When we consider the symmetric difference in terms of the regions mentioned, both groups of calculations \( (A + B) + C \) and \( A + (B + C) \) include regions \(1, 3, 5,\) and \(7\). This consistency demonstrates how the associative property guarantees reliability, even when the set operations become complex.
Math operations like addition and multiplication also exhibit this property, ensuring predictable and stable results no matter how terms are grouped within the operation.
Commutative Property
The commutative property highlights that the order of operations does not affect the result. In terms of our symmetric difference operation, it holds that \( A + B = B + A \) for any two sets \( A \) and \( B \).
Commutativity ensures that even if we swap the sets in the operation, we end up with the same result. This can be seen visually in symmetric differences, where the regions remain identical, regardless of the order of the two sets involved.
Commutative property is a fundamental feature across various operations like addition and multiplication in mathematics. It simplifies calculations and proofs since reversing the order of operations doesn't necessitate recalculating or adjusting results.
In our symmetric difference example, such consistency makes building operation tables and verifying results easier, reinforcing the mathematical structure we rely on for computations and theoretical propositions.
Commutativity ensures that even if we swap the sets in the operation, we end up with the same result. This can be seen visually in symmetric differences, where the regions remain identical, regardless of the order of the two sets involved.
Commutative property is a fundamental feature across various operations like addition and multiplication in mathematics. It simplifies calculations and proofs since reversing the order of operations doesn't necessitate recalculating or adjusting results.
In our symmetric difference example, such consistency makes building operation tables and verifying results easier, reinforcing the mathematical structure we rely on for computations and theoretical propositions.
Other exercises in this chapter
Problem 2
Prove that each of the following sets, with the indicated operation, is an abelian group. $$ x * y=\frac{x y}{2}, \text { on the set }\\{x \in \mathbb{R}: x \ne
View solution Problem 3
Show that \(\left(a_{1}, \ldots, a_{n}\right)+\left[\left(b_{1}, \ldots, b_{n}\right)+\left(c_{1}, \ldots, c_{n}\right)\right]=\left[\left(a_{1}, \ldots, a_{n}\
View solution Problem 3
Prove that each of the following sets, with the indicated operation, is an abelian group. $$ x * y=x+y+x y, \text { on the set }\\{x \in \mathbb{R}: x \neq-1\\}
View solution Problem 4
The symbol \(\mathbb{R} \times \mathbb{R}\) represents the set of all ordered pairs \((x, y)\) of real numbers. \(\mathbb{R} \times \mathbb{R}\) may therefore b
View solution