Problem 1 - Applied Discrete Structures

Question

For each of the subsets of the indicated monoid, determine whether the subset is a submonoid. (a) \(S_{1}=\\{0,2,4,6\\}\) and \(S_{2}=\\{1,3,5,7\\}\) in \(\left[\mathbb{Z}_{8} ; \times_{8}\right]\) (b) \(\left\\{f \in \mathbb{N}^{\mathbb{N}}: f(n) \leqslant n, \forall n \in \mathbb{N}\right\\}\) and \(\left\\{f \in \mathbb{N}^{\mathbb{N}}: f(1)=2\right\\}\) in the monoid \(\left[\mathbb{N}^{\mathbb{N}} ; \circ\right]\) (c) \(\\{A \subseteq \mathbb{Z} \mid A\) is finite \(\\}\) and \(\left\\{A \subseteq \mathbb{Z} \mid A^{c}\right.\) is finite \(\\}\) in \([\mathcal{P}(\mathbb{Z}) ; \cup]\).

Step-by-Step Solution

Answer

(a) \(S_1\) is not, \(S_2\) is a submonoid. (b) First set is, second is not a submonoid. (c) First is, second is not a submonoid.

Step 1: Understanding the Monoid

A monoid is a set equipped with an associative binary operation and an identity element. In this exercise, determine whether the given subsets are submonoids of the specified monoids.

Step 2: Part (a) Analysis - Submonoids in \(\left[\mathbb{Z}_{8} ; \times_{8}\right]\)

Check if subsets \(S_{1}=\{0,2,4,6\}\) and \(S_{2}=\{1,3,5,7\}\) are closed under multiplication modulo 8 and contain the identity element (1 in this case). - For \(S_1\): - Elements: 0, 2, 4, 6. - 1 is not in \(S_1\), so it's not a submonoid.- For \(S_2\): - Check closure: \(1 \times_{8} 3 = 3, \) included; similarly for other combinations like \(3 \times_{8} 5 = 7\), satisfied. - Contains 1 (identity under multiplication modulo 8), hence, \(S_2\) is a submonoid.

Step 3: Part (b) Analysis - Submonoids in \(\left[\mathbb{N}^{\mathbb{N}} ; \circ\right]\)

Check if subsets \(\{f \in \mathbb{N}^{\mathbb{N}}: f(n) \leqslant n, \forall n \in \mathbb{N}\}\) and \(\{f \in \mathbb{N}^{\mathbb{N}}: f(1)=2\}\) are closed under function composition and contain the identity function \(id(n) = n\). - For the first subset, closure holds since composing \(f(n) \leq n\) with another function \(g(n) \leq n\) gives a function \((f \circ g)(n) \leq n\). Identity function \(id(n) = n\) included.- For the second subset, not every function \(g\) satisfies \((f \circ g)(1)=2\)for \(f(1) = 2\), hence not closed under composition.

Step 4: Part (c) Analysis - Submonoids in \([\mathcal{P}(\mathbb{Z}) ; \cup]\)

Check subsets \(\{A \subseteq \mathbb{Z} \mid A\) is finite \(\}\) and \(\{A \subseteq \mathbb{Z} \mid A^{c}\) is finite \(\}\) for closure under union and presence of the empty set (identity for union).- First subset: Closed under finite union since the union of finite sets is finite and includes the empty set. Valid submonoid.- Second subset: Union of complements of finite sets \(A^c\) may not remain finite. For example, \(A \cup B = \mathbb{Z}\) could result in \((A \cup B)^c = \emptyset\) being finite, but the individual complements may not ensure closure. Hence, not a submonoid.

Key Concepts

Monoid

A monoid is a fascinating concept in abstract algebra. It refers to a set that is paired with an operation. This operation must satisfy two main conditions: it must be associative, and there must be an identity element within the set.

• Associativity means that changing the grouping of operations does not change the result. For example, if you have three elements \(a\), \(b\), and \(c\) and a binary operation \(*\), the equation \((a * b) * c = a * (b * c)\) holds.
• The identity element is an element in the set such that the operation on any element combined with it leaves the other element unchanged. For instance, in the set of numbers under addition, the identity element is 0, since \(a + 0 = a\).

By ensuring these properties, a monoid forms a very structured environment which is essential in various areas like computer science and mathematical computations.

Binary Operation

In mathematics, a binary operation is a type of operation that combines two elements (from a set) to produce another element from the same set. Understanding this concept is crucial when studying structures like monoids.
Here are some common examples and properties:

• Addition and multiplication among numbers are typical examples.
• The operation must be defined for every pair of elements in the set.
• It usually needs to satisfy specific properties to fit within additional structures like groups or rings.

In a monoid, this operation must be associative. To identify a submonoid, we ensure that this operation, when applied to elements of a subset, results in an outcome still within the same subset.

Identity Element

The identity element is a special and critical component in algebraic structures such as monoids. It's the element which, when used in a binary operation with any other element from the set, leaves that element unchanged.
For instance:

• In a monoid with the set of integers under multiplication, the identity element is 1, because \(a \times 1 = a\) for any integer \(a\).
• In another monoid with functions under composition, the identity is the identity function \(id(x) = x\), since applying it to any function \(f(x)\) gives the same function \(f(x)\).

Recognizing the identity element is vital when determining if a subset is a submonoid, as it must include this element to uphold the monoid properties.

Function Composition

Function composition is the process of applying one function to the results of another. This forms a fundamental operation in algebraic structures like monoids where the set consists of functions.
Here's a breakdown:

• Given two functions \(f\) and \(g\), their composition \((f \circ g)(x)\) means first applying \(g\) to \(x\), then \(f\) to the result \(g(x)\).
• This operation is associative: \((f \circ (g \circ h)) = ((f \circ g) \circ h)\).

In determining submonoids, we check if composing any functions from a subset stays within that subset. The presence or absence of the identity function can also play a critical role in these determinations.