Problem 1 - Applied Discrete Structures

Question

Draw the transition dingrams for the machines of the following monoids: (a) \(\left[Z_{4} ;+_{4}\right]\) (b) The direct product of \(\left[\mathrm{Z}_{2} ; \mathrm{x}_{2}\right]\) with itself.

Step-by-Step Solution

Answer

Draw states and transitions based on addition or multiplication modulo 4/2.

Step 1: Understanding the problem

We are asked to draw transition diagrams for machines of two monoids, namely \( \left[Z_{4}; +_{4}\right] \) and the direct product of \( \left[Z_{2}; x_{2}\right] \) with itself. Monoid operations concern addition or multiplication, with operations performed modulo a certain number.

Step 2: Prepare for transition diagram of \( \left[Z_{4}; +_{4}\right] \)

This monoid contains elements \(\{0, 1, 2, 3\}\) with the operation being addition modulo 4. Each element will have transitions to all other elements based on addition and reducing modulo 4.

Step 3: Draw transition diagram for \( \left[Z_{4}; +_{4}\right] \)

Create states \(0, 1, 2, 3\). For each state, add a transition: \ - State 0 transitions to itself on input 0, to state 1 on input 1, to state 2 on input 2, to state 3 on input 3.\ - State 1 transitions to state 1 on input 0, state 2 on input 1, state 3 on input 2, and state 0 on input 3.\ - Continue this pattern for states 2 and 3. Label each transition based on the sum modulo 4.

Step 4: Define and prepare for transition diagram of \( \left[\mathrm{Z}_{2} ; \mathrm{x}_{2}\right] \times \left[\mathrm{Z}_{2} ; \mathrm{x}_{2}\right] \)

The direct product \( \left[\mathrm{Z}_{2} ; \mathrm{x}_{2}\right] \) with itself results in elements \(\{(0,0), (0,1), (1,0), (1,1)\}\). The operation is performed element-wise using multiplication modulo 2.

Step 5: Draw transition diagram for the direct product

Create states representing each element \((0,0), (0,1), (1,0), (1,1)\). For each element, calculate transitions: \ - For \((0,0)\), any operation will remain \((0,0)\) because multiplication by 0 results in 0.\ - \((1,0)\) transitions remain \((0,0)\) except for an input of \((1,1)\), which yields \((1,0)\). \ - Similarly compute for \((0,1)\) and \((1,1)\). Draw transitions based on calculated operations.

Key Concepts

Monoids

Monoids are foundational structures in algebra, often introduced in mathematics without context. They're essentially a set equipped with a binary operation that's closed, associative, and has an identity element. Let's break this down more:
- **Closure** means that when you perform the operation on any two elements of the set, the result is also in the set. Imagine a group of numbers where the outcome stays in the group.
- **Associativity** implies that changing the grouping of operations doesn't alter the outcome, (e.g., \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\)).
- **Identity** is an element in the set that doesn't change another element when used in the operation (like 0 in addition).
Consider the set of numbers \(Z_4\) under addition modulo 4. The set \(\{0, 1, 2, 3\}\) with addition forms a monoid because it meets all these conditions. This operation cycle under the modulo condition ensures that each outcome fits back into the set, completing the monoid description.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around once they reach a certain value, known as the modulus. It functions like the hours on a clock, cycling back after reaching a point.
Using modular arithmetic, \( a \mod n \) gives the remainder when \(a\) is divided by \(n\).
For our exercise, working with \(Z_4\) involves performing addition modulo 4:

• This means 3 added to 2 becomes 1 (\