Problem 26
Question
Let \(U=\\{\mathrm{a}, \ldots, \mathrm{g}\\} .\) Define the characteristic function \(h\) of each set. $$\\{\mathrm{a}, \mathrm{c}, \mathrm{d}, \mathrm{f}, \mathrm{g}\\}$$
Step-by-Step Solution
Verified Answer
The characteristic function \(h(x)\) of the given set \(\{a, c, d, f, g\}\) is defined as:
\[h(x) = \begin{cases}
1 & \text{if}\ x = a, c, d, f, g \\
0 & \text{otherwise}
\end{cases}\]
1Step 1: Define the characteristic function
The characteristic function h(x) of a specific set (A) is defined as:
\[h(x) = \begin{cases}
1 & \text{if}\ x \in A \\
0 & \text{if}\ x \notin A \end{cases}\]
Now we need to find the characteristic function for the given set {a, c, d, f, g}.
2Step 2: Apply the characteristic function to each element
We will now apply the function h(x) to each element in the universal set U, and check if the element is present in the given set {a, c, d, f, g}:
\(h(a) = 1\) (as a is present in the given set)
\(h(b) = 0\) (as b is not present in the given set)
\(h(c) = 1\) (as c is present in the given set)
\(h(d) = 1\) (as d is present in the given set)
\(h(e) = 0\) (as e is not present in the given set)
\(h(f) = 1\) (as f is present in the given set)
\(h(g) = 1\) (as g is present in the given set)
3Step 3: Write down the complete characteristic function
From the previous step, we can now write the complete characteristic function h(x) for the given set {a, c, d, f, g} as:
\[h(x) = \begin{cases}
1 & \text{if}\ x = a, c, d, f, g \\
0 & \text{otherwise}
\end{cases}\]
Key Concepts
Discrete MathematicsSet TheoryUniversal SetElement of a Set
Discrete Mathematics
Discrete mathematics is a branch of mathematics that deals with discrete elements that are distinct and separate. This field includes the study of mathematical structures that are fundamentally discrete rather than continuous. Key topics within discrete mathematics include algorithms, combinatorics, graph theory, and logic.
One of the foundational aspects of discrete mathematics is the manner in which it deals with data sets and elements. The problems in discrete mathematics often involve counting, logic, or arrangement where set theory plays a pivotal role. In the exercise provided, the concept of a characteristic function is a discrete function that indicates the presence or absence of an element in a set, showing how discrete math can handle and simplify information for analysis or further computation.
One of the foundational aspects of discrete mathematics is the manner in which it deals with data sets and elements. The problems in discrete mathematics often involve counting, logic, or arrangement where set theory plays a pivotal role. In the exercise provided, the concept of a characteristic function is a discrete function that indicates the presence or absence of an element in a set, showing how discrete math can handle and simplify information for analysis or further computation.
Set Theory
Set theory is the mathematical study of collections of objects, which are known as sets. It provides a foundational system for nearly all of mathematics and it deals with the properties and relationships of sets. A set can be described as a grouping of elements that are considered a whole. Sets are one of the most fundamental concepts in mathematics and are used to define and establish the concept of numbers, relations, functions, and more.
In the context of the exercise, the characteristic function is a tool from set theory that helps us understand which elements belong to a certain set and which do not. Set theory helps define the characteristic function, which is a simple way to express sets using binary notation (0s and 1s), with 1 representing an element's presence within the set and 0 its absence.
In the context of the exercise, the characteristic function is a tool from set theory that helps us understand which elements belong to a certain set and which do not. Set theory helps define the characteristic function, which is a simple way to express sets using binary notation (0s and 1s), with 1 representing an element's presence within the set and 0 its absence.
Universal Set
In set theory, the universal set (denoted as U) is the set that contains all objects and elements under consideration for a particular discussion or problem. Every other set in that context is a subset of the universal set. It's the 'universe' of relevant elements.
In the exercise, the universal set U is defined as \(U = \{a, b, c, d, e, f, g\}\). When determining the characteristic function for a particular set, we consider the elements of this universal set to decide the binary outcome for each element (0 or 1). By referencing the universal set, we can ascertain the comprehensiveness of the characteristic function as it must account for every element within the universal set.
In the exercise, the universal set U is defined as \(U = \{a, b, c, d, e, f, g\}\). When determining the characteristic function for a particular set, we consider the elements of this universal set to decide the binary outcome for each element (0 or 1). By referencing the universal set, we can ascertain the comprehensiveness of the characteristic function as it must account for every element within the universal set.
Element of a Set
An element of a set is any one of the distinct objects that make up that set. The concept of an element is fundamental to the understanding of sets; when we say that an object is an 'element of a set,' we are stating that the object belongs to the set. In notation, if we have an element a and a set A, we represent 'a is an element of A' as \(a \in A\).
The characteristic function described in the exercise identifies whether a particular element is within the set in question. If we consider the element 'a' from the universal set U and the set {a, c, d, f, g}, we can see that 'a' is an element of this set, leading the characteristic function to assign a 1 to it (\(h(a) = 1\)). This binary method provides a clear and concise way to represent elements' membership status in regards to a specific set.
The characteristic function described in the exercise identifies whether a particular element is within the set in question. If we consider the element 'a' from the universal set U and the set {a, c, d, f, g}, we can see that 'a' is an element of this set, leading the characteristic function to assign a 1 to it (\(h(a) = 1\)). This binary method provides a clear and concise way to represent elements' membership status in regards to a specific set.
Other exercises in this chapter
Problem 26
Let \(f: \mathbf{Z} \times \mathbf{Z} \rightarrow \mathbf{Z}\) defined by \(f(x, y)=2 x+3 y-6 x y .\) Compute the following. $$f(-3,0)$$
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Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll} 1 & \text { if } i=j \\ 0 & \text { otherwise } \end{
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Determine if the functions are bijective. If they are not bijective, explain why. The ORD function on ASCII.
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Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }
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