Algebraic Structures Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Sofia Scott (Solvelet student)

For the set

S = \{0, 1, 2, 3\}

under addition modulo 4, verify if

S

forms a group.

Solution

Step 1: Check closure. For all

a, b \in S

a + b \mod 4 \in S

. Verify by constructing the addition table modulo 4: \begin{center} \begin{tabular}{c|cccc} + & 0 & 1 & 2 & 3 \ \hline 0 & 0 & 1 & 2 & 3 \ 1 & 1 & 2 & 3 & 0 \ 2 & 2 & 3 & 0 & 1 \ 3 & 3 & 0 & 1 & 2 \ \end{tabular} \end{center} Step 2: Check associativity. Addition modulo 4 is associative because regular addition is associative and taking modulo does not affect this property. Step 3: Identify the identity element. The identity element

e

must satisfy

a + e \equiv a \mod 4

. Clearly,

0

is the identity element. Step 4: Check for inverses. For each

a \in S

, find an element

b \in S

such that

a + b \equiv 0 \mod 4

\begin{aligned} 0 + 0 &\equiv 0 \mod 4 &\rightarrow 0 \text{ is its own inverse}\ 1 + 3 &\equiv 0 \mod 4 &\rightarrow 1 \text{ and } 3 \text{ are inverses}\ 2 + 2 &\equiv 0 \mod 4 &\rightarrow 2 \text{ is its own inverse} \end{aligned}

Step 5: Conclusion. Since

S

under addition modulo 4 is closed, associative, has an identity element, and every element has an inverse,

S

forms a group. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Jacob Robinson on Solvelet

1. Show that the set of integers under addition forms a group 2. Prove that the set of all

2 \times 2

invertible matrices with matrix multiplication forms a group.,

DefinitionIn mathematical language, algebraic structures are composed of sets together with one or more operations defined on them, satisfies certain axioms. These structures can be groups, rings, fields, or vector spaces, each endowed with formal definitions of the operations that hold for that class of objects. Example: The set of integers along with the associated operations of addition and multiplication forms a ring in which addition is associative and multiplication distributes over addition. Solve & Learn Algebraic Structures solved Step by step answers. Automatically Generated, Understand and explanation of Algebraic Structures calculator on SolveletAI.

Related Calculators

Integral Calculator Radius and Interval of Convergence Calculator Curve Sketching Calculator Proving trigonometric identities Calculator

Need topic explanation ? Get video explanation