Problem 24 - Discrete Mathematics with Applications

Question

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{j=1}^{6} \sum_{i=1}^{5}(2 i+3 i) $$

Step-by-Step Solution

Answer

The short answer for evaluating the sum \(\sum_{j=1}^{6} \sum_{i=1}^{5}(2i + 3i)\) using Kronecker's delta is 275.

Step 1: Break the given sum into individual sums

First, we need to express the given sum in terms of Kronecker's delta. We have the following sum to evaluate: $$ \sum_{j=1}^{6} \sum_{i=1}^{5}(2i+3i) $$ Since Kronecker's delta is involved, rewrite the sum using \(\delta_{ij}\). We can write it as: $$ \sum_{j=1}^{6} \sum_{i=1}^{5}(2i\delta_{ij}+3i\delta_{ij}) $$

Step 2: Evaluate each sum individually

Now, let's evaluate each sum using the definition of Kronecker's delta. Evaluate the first sum with \(2i\delta_{ij}\): $$ \begin{aligned} \sum_{j=1}^{6} \sum_{i=1}^{5} 2i\delta_{ij} &= \sum_{j=1}^{6} \left(2\cdot1\delta_{1j} + 2\cdot2\delta_{2j} + 2\cdot3\delta_{3j} + 2\cdot4\delta_{4j} + 2\cdot5\delta_{5j}\right) \\ &= 2\left(\sum_{j=1}^{6}\delta_{1j} + 2\sum_{j=1}^{6}\delta_{2j}+ 3\sum_{j=1}^{6}\delta_{3j} + 4\sum_{j=1}^{6}\delta_{4j}+ 5\sum_{j=1}^{6}\delta_{5j}\right) \end{aligned} $$ Evaluate the second sum with \(3i\delta_{ij}\): $$ \begin{aligned} \sum_{j=1}^{6} \sum_{i=1}^{5} 3i\delta_{ij} &= \sum_{j=1}^{6} \left(3\cdot1\delta_{1j} + 3\cdot2\delta_{2j} + 3\cdot3\delta_{3j} + 3\cdot4\delta_{4j} + 3\cdot5\delta_{5j}\right) \\ &= 3\left(\sum_{j=1}^{6}\delta_{1j} + 2\sum_{j=1}^{6}\delta_{2j}+ 3\sum_{j=1}^{6}\delta_{3j} + 4\sum_{j=1}^{6}\delta_{4j}+ 5\sum_{j=1}^{6}\delta_{5j}\right) \end{aligned} $$

Step 3: Add both sums by applying Kronecker's delta definition

Now, let's add both sums by applying Kronecker's delta definition: $$ \begin{aligned} \sum_{j=1}^{6} \sum_{i=1}^{5}(2i\delta_{ij}+3i\delta_{ij}) &= 2\left(\sum_{j=1}^{6}\delta_{1j} + 2\sum_{j=1}^{6}\delta_{2j}+ 3\sum_{j=1}^{6}\delta_{3j} + 4\sum_{j=1}^{6}\delta_{4j}+ 5\sum_{j=1}^{6}\delta_{5j}\right) \\ &+ 3\left(\sum_{j=1}^{6}\delta_{1j} + 2\sum_{j=1}^{6}\delta_{2j}+ 3\sum_{j=1}^{6}\delta_{3j} + 4\sum_{j=1}^{6}\delta_{4j}+ 5\sum_{j=1}^{6}\delta_{5j}\right) \end{aligned} $$ We know that \(\delta_{ij} = 1\) if \(i = j\), and \(\delta_{ij} = 0\) if \(i \neq j\). So, the above sum becomes: $$ \begin{aligned} &= 2\left(1 + 4 + 9 + 16 + 25\right) + 3\left(1 + 4 + 9 + 16 + 25\right) \\ &= 2(55) + 3(55) \\ &= 5(55) \end{aligned} $$ Therefore, the sum is: $$ \sum_{j=1}^{6} \sum_{i=1}^{5}(2i\delta_{ij} + 3i\delta_{ij}) = 5(55) = 275 $$

Key Concepts

Discrete Mathematics

In the fascinating world of mathematics, discrete mathematics is concerned with the study and analysis of discrete objects. Unlike continuous mathematics, where information can change continuously, discrete mathematics deals with objects that can only take on distinct, separated values. This field includes a variety of topics such as logic, set theory, graph theory, and combinatorics. One of its vital components is the concept of Kronecker's delta, which is used in summation and series. Kronecker's delta helps in simplifying the process of adding up series by using the trick of turning non-qual terms essentially into zeros and keeping the equal terms as ones, thereby managing the summation process efficiently.

Summation Notation

Summation notation is a concise way to represent the addition of a sequence of numbers using the Greek letter sigma (\f\(\f\)), followed by an expression for the terms to be summed. It allows mathematicians to write a long series of addition in a compact form. For example, the notation \f\(\f_j=1^n a_j\f\) means that we are adding up the terms \f\(a_1, a_2, ..., a_n\f\). Summation can be thought of as an instruction that tells you to add up a set of numbers, where the numbers are described by a pattern or a rule.

Double Summation

Double summation takes the concept of summation one step further by summing over two different indices. You can envision this as adding up elements in a two-dimensional array, much like filling up a bingo card vertically and horizontally. It is denoted using two sigma symbols—for instance, \f\(\f_{i=1}^m \f_{j=1}^n a_{ij}\f\). This instructs us to sum across one index while holding the other index constant and then sum the results across the other index. With the Kronecker's delta present, we simplify summations by eliminating terms unless the indices match, which streamlines the process considerably.

Mathematical Series

A mathematical series is the sum of the terms of a sequence of numbers. It allows us to sum infinitely many terms to figure out their total, which can be finite or infinite depending on the properties of the sequence. Series can be simple or quite complex, involving various functions and forms. When using the Kronecker's delta within series, as seen in the exercise provided, we leverage its value-altering feature based on the indices to selectively sum terms in a series. This selective summation cuts through the noise and focuses solely on the significant terms of the series, making the evaluation of the sum far more manageable.