Problem 4 - Applied Discrete Structures

Question

Verify the following properties for \(n=3\). (a) \(\sum_{i=1}^{n}\left(a_{i}+b_{i}\right)=\sum_{i=1}^{n} a_{i}+\sum_{i=1}^{n} b_{i}\) (b) \(c\left(\sum_{i=1}^{n} a_{i}\right)=\sum_{i=1}^{n} c a_{i}\)

Step-by-Step Solution

Answer

Both properties are verified for \( n=3 \).

Step 1: Understanding the Problem

We need to verify two properties for the case where \( n = 3 \). The first property involves the sum of two sequences, and the second property involves distributing a constant over a sum.

Step 2: Expanding the First Property

For (a), expand both sides of the equation when \(n=3\). The left side is: \( \sum_{i=1}^{3} (a_i + b_i) = (a_1 + b_1) + (a_2 + b_2) + (a_3 + b_3) \). The right side is: \( \sum_{i=1}^{3} a_i + \sum_{i=1}^{3} b_i = (a_1 + a_2 + a_3) + (b_1 + b_2 + b_3) \).

Step 3: Evaluating the Expanded Sums

Notice that the left side: \((a_1 + b_1) + (a_2 + b_2) + (a_3 + b_3)\) can be rearranged as: \((a_1 + a_2 + a_3) + (b_1 + b_2 + b_3)\), which precisely matches the right side. Hence, (a) is verified.

Step 4: Expanding the Second Property

For (b), expand both sides for \(n=3\). The left side is: \( c( \sum_{i=1}^{3} a_i ) = c(a_1 + a_2 + a_3) \). The right side is: \( \sum_{i=1}^{3} ca_i = ca_1 + ca_2 + ca_3 \).

Step 5: Simplifying the Second Property

The left side \( c(a_1 + a_2 + a_3) \) can be distributed to \( ca_1 + ca_2 + ca_3 \), which is identical to the right side. Therefore, (b) is verified.

Key Concepts

Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement. In the realm of discrete mathematics, a proof involves confirming the validity of properties or theorems based on established mathematical principles.
To construct a proof:

• First, understand the problem and identify what statement needs to be proven.
• Break down the statement logically, using known formulas and concepts.
• Use logical reasoning to show that the proposed statement follows from existing truths.
• Finally, conclude by verifying that each step logically leads to the next, thereby confirming the statement as true.

In the exercise provided, mathematical proofs are used to confirm two properties of summations with a simple case when \( n=3 \). Each proof demonstrates that both sides of the given equations are indeed equal when expanded and rearranged accordingly.
By mastering mathematical proofs, students can validate properties confidently and develop critical thinking skills essential for coursework in discrete mathematics and beyond.

Summation Notation

Summation notation is a convenient way to express the sum of a sequence of terms. It is generally represented with the Greek letter \( \Sigma \). This notation simplifies the representation of adding multiple terms, especially when dealing with long sequences.
Key features of summation notation:

• It is expressed as \( \sum_{i=1}^{n} a_i \), where \( i \) is the index of summation, starting at 1 and ending at \( n \).
• \( a_i \) represents each term in the sequence to be summed.
• The notation allows for compact expression of adding up similar terms without writing them out in full.

In our exercise, summation notation is used to describe the sum of sequences \( a_i \) and \( b_i \). The notation helps to easily manipulate and verify properties of sequences, such as distributing and re-grouping terms. Understanding and utilizing summation notation is a crucial skill in efficiently solving complex sequences and series problems.

Distributive Property

The distributive property is a fundamental concept in mathematics that explains how multiplication interacts with addition. Simply put, it allows you to multiply a sum by a number by multiplying each addend separately and then adding the products. The distributive property can be expressed as:

• \( c(a + b) = ca + cb \)
• This property is essential for simplifying expressions and solving equations.

In the context of the given exercise, the distributive property is applied to confirm that multiplying a constant with a sum (\( c(\sum_{i=1}^{n} a_i) \)) is equivalent to adding each term in the sum after it has been multiplied by the constant (\( \sum_{i=1}^{n} ca_i \)).
By applying the distributive property, you can efficiently simplify expressions and verify identities involving sums and multiplicative constants. Understanding this key concept provides a foundation for much of algebra and further studies in mathematics.

Sequence and Series

Sequences and series are foundational concepts in discrete mathematics that involve the arrangement and aggregation of numbers. A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.
Key distinctions and definitions:

• A sequence is written as \( a_1, a_2, a_3, \ldots, a_n \).
• A series, when summing a sequence, is denoted by \( \sum_{i=1}^{n} a_i \).
• Understanding both allows for manipulation of numerical patterns and capturing the essence of mathematical growth or trends.

In this exercise, we explore properties related to the summation of sequences (the series). The first property to verify involves the summation of addition within sequences, while the second explores distributing constants across sums.
Mastering sequences and series is crucial for students studying discrete mathematics because they underpin crucial mathematical processes and provide insights into how numerical trends evolve.