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Chapter 5: Problem 13

For \(A, B, C \subseteq ?\), prove that \(A \times(B-C)=(A \times B)-(A \times C)\).

Short Answer

Expert verified
The given equation \(A \times(B-C)=(A \times B)-(A \times C)\) can be proven by demonstrating that any arbitrary element in the left-hand side set is also an element of the right-hand side set, and vice versa. This shows that the two sets are equal. Note that the key to the proof is understanding the operations of Cartesian product and set difference.

Step by step solution

01

Understanding what the problem is asking for

First, it's important to understand what needs to be proven. The equation involves the operations of Cartesian product and set difference, denoted by \(\times\) and - respectively. The Cartesian product is a set formed by all possible ordered pairs combining elements from two other sets, and the set difference is simply all elements in one set that aren't in another. The equation is saying that if you take the Cartesian product of a set A and the difference between two sets B and C, then it is equivalent to taking the Cartesian product of A with B and then subtracting the Cartesian product of A with C.
02

Writing out explicit definitions

Next, one needs to write out the explicit definitions of each term in the equation. This will make it clear what each operation is actually doing. So, \(A \times(B-C)\) refers to all ordered pairs (a, b) such that a is an element of A and b is an element of B but not of C, and \(A \times B\) and \(A \times C\) refer to all ordered pairs (a, b) and (a, c) such that a is an element of A and b and c are elements of B and C, respectively.
03

Proving the equation

Now, one starts the proof by picking an arbitrary element from the left side of the equation and shows that it must be an element of the right side, and vice-versa. If ((a, b) is an element of \(A \times(B-C)\)), this means that a is an element of A and b is an element of B but not of C. This implies that (a, b) is an element of \(A \times B\), but (a, b) is not an element of \(A \times C\), since b is not an element of C. Hence, (a, b) is an element of \((A \times B)-(A \times C)\). Therefore, \(A \times(B-C) \subseteq (A \times B)-(A \times C)\). Using similar argument it can be shown that, \(A \times (B-C) \supseteq (A \times B)-(A \times C)\). Hence, one can conclude that \(A \times (B-C) = (A \times B)-(A \times C)\).

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Key Concepts

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

Cartesian Product
The Cartesian Product is a fundamental concept in set theory, crucial for understanding many mathematical structures. Given two sets, say \( A \) and \( B \), the Cartesian Product is denoted by \( A \times B \). It consists of all possible ordered pairs where the first element is from set \( A \) and the second element is from set \( B \).
For example, if \( A = \{1, 2\} \) and \( B = \{x, y\} \), the Cartesian Product \( A \times B \) would be \( \{ (1, x), (1, y), (2, x), (2, y) \} \). This is a set of ordered pairs.

The Cartesian Product is especially important because it allows for the combination of different sets to explore relationships or define multidimensional spaces.
  • Ordered pairs: Each combination consists of two elements, the first from the first set, and the second from the second set.
  • Dimensionality: The product of two sets creates a rectangular grid-like structure in a two-dimensional space.
  • Applications: This concept is used in databases, cross-tabulation, and defining spatial dimensions.
Understanding the Cartesian Product helps in grasping more complex structures like matrices and relations.
Set Difference
The Set Difference is another fundamental operation in set theory. It is used to determine the elements that belong to one set but not to another. This operation is usually denoted by the minus sign (-).
For sets \( B \) and \( C \), the set difference \( B - C \) consists of all elements that are in \( B \) but not in \( C \).

For example, if \( B = \{1, 2, 3\} \) and \( C = \{2, 3, 4\} \), then \( B - C \) would be \( \{1\} \) because 1 is in \( B \) but not in \( C \).
  • Non-commutativity: The operation \( B - C \) does not equal \( C - B \), due to the specific membership of elements.
  • Subset relationship: The set difference can be thought of as subtracting set \( C \) from set \( B \).
  • Applications: This operation is widely used in computing, database management and logical expressions.
Understanding the set difference helps highlight key elements that distinguish between sets, which is vital for problem-solving and logical deductions.
Ordered Pairs
In the context of set theory, an ordered pair refers to a specific sequence of two elements. These are denoted by \( (a, b) \), where \( a \) is from the first set and \( b \) from the second set.
One of the significant attributes of ordered pairs is that the order matters 鈥� \( (a, b) \) is not the same as \( (b, a) \) unless \( a = b \).

Ordered pairs are foundational for defining relations and mappings in mathematics and are also used to form the building blocks of Cartesian Products.
  • Uniqueness of order: Changing the order of elements creates a distinct ordered pair.
  • Identity: For an ordered pair to be equal to another, each respective element must be equal.
  • Applications: Ordered pairs are used in coordinate systems, functions, and data representation in many fields.
Grasping the concept of ordered pairs is essential for understanding operations in set theory and beyond in areas such as algebra, geometry, and computer science.

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Most popular questions from this chapter

If \(f: \mathbf{R} \rightarrow \mathbf{R}\) with \(f(x)=x^{n}\), for which \(n \in \mathbf{Z}^{+}\)is \(f\) invertible?

With \(A=\\{x, y, z\\}\), let \(f, g: A \rightarrow A\) be given by \(f=\\{(x, y),(y, z),(z, x)\\}, g=\\{(x, y),(y, x),(z, z)\\}\). Determine each of the following: \(f \circ g, g \circ f, f^{-1}, g^{-1},(g \circ f)^{-1}\), \(f^{-1} \circ g^{-1}\), and \(g^{-1} \circ f^{-1}\).

The following Pascal function and program segment are used to evaluate the polynomial $$ 8-10 x+7 x^{2}-2 x^{3}+3 x^{4}+12 x^{5} . $$ The function Power is used to determine the value of \(r^{k}\) for the real variable \(r\) and integer variable \(k\), where \(k>0\). In the program segment the variable Sum is a real variable while the variable \(j\) is an integer variable, and prior to execution a value is assigned to the (previously defined) real variable \(x\). Also, the (previously defined) array variable \(a\) has been assigned the integer components $$ a[0]=8, \quad a[1]=-10, \quad a[2]=7, \quad a[3]=-2, \quad a[4]=3, \quad \text { and } \quad a[5]=12 $$ Function Power (r; real; k: integer): real; Var 1: 1nteger; product: real; Begin product : = 1; For : : = 1 to k do product : = product * r; Power := product End; Sum : \(=\) a[0]; For \(j:=1\) to 5 do Sum : = Sum + a[j] * Power \((x, j) ;\) a) How many additions take place in the evaluation of the given polynomial? How many multiplications? b) How many additions and how many multiplications take place if we adjust the input and program segment to deal with the polynomial $$ c_{0}+c_{1} x+c_{2} x^{2}+c_{3} x^{3}+\cdots+c_{n-1} x^{n-1}+c_{n} x^{n}, $$ where \(c_{0}, c_{1}, c_{2}, c_{3}, \ldots, c_{n-1}, c_{n}\) are integers and \(n\) is a positive integer?

Let \(A_{1}, A\) and \(B\) be sets with \(\\{1,2,3,4,5\\}=\) \(A_{1} \subset A, B=\\{s, t, u, v, w, x\\}\), and \(f: A_{1} \rightarrow B\). If \(f\) can be extended to \(A\) in 216 ways, what is \(|A|\) ?

Let \(A=\\{1,2,3,4,5\\}\) and \(B=\\{t, u, v, w, x, y, z\\}\). (a) If a function \(f: A \rightarrow B\) is randomly generated, what is the probability that it is one-to-one? (b) Write a computer program (or develop an algorithm) to generate random functions \(f: A \rightarrow B\) and have the program print out how many functions it generates until it generates one that is one-to-one.
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