Problem 13 - Applied Discrete Structures

Question

(a) For any ring \([R ;+, \cdot]\), expand \((a+b)(c+d)\) for \(a, b, c, d \in R\). (b) If \(R\) is commutative, prove that \((a+b)^{2}=a^{2}+2 a b+b^{2}\) for all \(a, b \in R\)

Step-by-Step Solution

Answer

(a) \((a+b)(c+d) = ac + ad + bc + bd\). (b) In a commutative ring, \((a+b)^2 = a^2 + 2ab + b^2\).

Step 1: Expanding the Product Expression

To expand the expression \((a+b)(c+d)\), we apply the distributive property. We distribute \(a+b\) over \(c+d\) which results in two separate operations: \((a+b) \cdot c + (a+b) \cdot d\). This further expands as follows:1. First, distribute \(a\): \(a \cdot c + a \cdot d\).2. Then, distribute \(b\): \(b \cdot c + b \cdot d\).Thus, combining these, the expanded form is \((a+b)(c+d) = a \cdot c + a \cdot d + b \cdot c + b \cdot d\).

Step 2: Expanding a Square Expression

Firstly, recall the expression \((a+b)^2\) is equivalent to \((a+b)(a+b)\). Using the expansion derived previously, distribute \((a+b)\) over itself:1. First, distribute \(a\): \(a \cdot a + a \cdot b\).2. Then, distribute \(b\): \(b \cdot a + b \cdot b\).This results in: \(a^2 + ab + ba + b^2\).

Step 3: Simplifying Using Commativity

In a commutative ring, we have \(ab = ba\). Thus, from the expansion \(a^2 + ab + ba + b^2\), substitute \(ba\) with \(ab\):- This gives \(a^2 + ab + ab + b^2\), which simplifies to \(a^2 + 2ab + b^2\).- Therefore, in a commutative ring, \((a+b)^2 = a^2 + 2ab + b^2\).

Key Concepts

Distributive Property

The distributive property is one of the cornerstones of algebra and plays a crucial role in various algebraic structures, including rings. In essence, it connects addition and multiplication operations to allow for the expansion of expressions. The property states: for any elements \(a, b, c\) in a ring \(R\), the following holds: \(a \cdot (b + c) = a \cdot b + a \cdot c\). In simpler terms, multiplication is distributed over addition.When expanding an expression like \((a+b)(c+d)\) in a ring, the distributive property is applied twice.

• First, treat \((a+b)\) as a single unit that multiplies with each component of \((c+d)\). This gives us \((a+b) \cdot c + (a+b) \cdot d\).
• Next, distribute each component individually: \(a \cdot c + a \cdot d\) and \(b \cdot c + b \cdot d\).
• Combine these to achieve the final expanded form: \(a \cdot c + a \cdot d + b \cdot c + b \cdot d\).

This method highlights how the distributive property helps simplify complex expressions by breaking them down into smaller, more manageable parts.

Commutative Ring

A commutative ring is a special type of ring where the multiplication operation is commutative. This means that the order in which two elements are multiplied does not affect the outcome. Mathematically, for elements \(a, b\) in a commutative ring \(R\), it holds that \(a \cdot b = b \cdot a\).This property is significant because it simplifies expression manipulations. When proving identities in a commutative ring, like \[ (a+b)^2 = a^2 + 2ab + b^2 \], the commutativity allows rearrangement of terms. During expansion, each occurrence of \(ab\) is interchangeable with \(ba\), effectively doubling the term \(ab\) here.

• Starting with the expanded form: \(a^2 + ab + ba + b^2\).
• Replace \(ba\) with \(ab\) to get \(a^2 + ab + ab + b^2\).
• This simplifies to \(a^2 + 2ab + b^2\), clearly showcasing the beauty of commutativity.

Thus, understanding the properties of a commutative ring can make algebraic proofs and simplifications more intuitive and systematic.

Expression Expansion

Expression expansion is a systematic approach used to transform an expression into a longer, more detailed form using properties of algebra, primarily the distributive property. This process allows complex expressions to be tackled more easily, as they are broken down into simpler parts.In studying how expressions expand, consider the expression \((a+b)^2\). The key here is realizing that this is equivalent to \((a+b)(a+b)\), meaning we treat it like the product of two binomials. Here's how expansion works step-by-step:

• Apply the distributive property to the first pair \((a+b)(a+b)\), leading to terms: \(a \cdot a + a \cdot b\).
• Next, distribute for the second part, resulting in: \(b \cdot a + b \cdot b\).
• These individual products are combined to yield: \(a^2 + ab + ba + b^2\).

Expression expansion helps in recognizing patterns and simplifies further manipulation or solving of algebraic expressions. Once expanded, further properties, such as commutativity, can aid in the simplification process, leading to results that are easier to interpret and apply.