Problem 23 - 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 \((1823-1891) . ]\) $$ \sum_{i=1}^{3} \sum_{j=1}^{i}(j+3) $$

Step-by-Step Solution

Answer

The short answer to the question is: The value of the given expression with nested summations is \(\boxed{28}\).

Step 1: Understand the Kronecker's delta

The given Kronecker's delta is represented by the symbol \(\delta_{ij}\), and it has two possible values: 1 if the indices i and j are equal, and 0 otherwise. It is a useful tool in working with indices and sums, though it does not appear directly in the expression to be evaluated.

Step 2: Analyze the given expression

The expression to evaluate is: $$ \sum_{i=1}^{3} \sum_{j=1}^{i}(j+3) $$ This expression contains nested summations, with the outer sum running over i from 1 to 3, and the inner sum running over j from 1 to i.

Step 3: Evaluate the inner sum

We start by evaluating the inner sum for each value of i. We have: For i = 1: $$ \sum_{j=1}^{1}(j+3) = (1 + 3) = 4 $$ For i = 2: $$ \sum_{j=1}^{2}(j+3) = (1 + 3) + (2 + 3) = 4 + 5 = 9 $$ For i = 3: $$ \sum_{j=1}^{3}(j+3) = (1 + 3) + (2 + 3) + (3 + 3) = 4 + 5 + 6 = 15 $$

Step 4: Evaluate the outer sum

Now we proceed to evaluate the outer sum: $$ \sum_{i=1}^{3} \sum_{j=1}^{i}(j+3) = 4 + 9 + 15 = 28 $$ Hence, the value of the given expression is 28.

Key Concepts

Nested Summations

Nested summations are a type of mathematical expression found frequently in discrete mathematics, where one summation is placed inside another. Think of it as a way to add up a series of numbers that are themselves sums. For our purposes, let's consider the textbook problem where we're dealing with a sum within a sum, indicated by the expression \[ \sum_{i=1}^{3} \sum_{j=1}^{i}(j+3) \].

To break this down, start by evaluating the inner sum, which can be thought of as the 'inner loop' in a programming context. In our example, for each fixed value of the outer variable 'i', we calculate the total of the inner variable 'j', plus 3. This process is repeated for each value that 'i' can take on. The key to mastering nested summations is to perform the calculation step by step, starting from the innermost sum and working your way outward.

By isolating each layer of the summation, the task becomes more manageable. It's similar to unpacking a set of Russian nesting dolls; each summation is considered individually, and when combined, they construct the entire expression.

Discrete Mathematics

Discrete mathematics is a branch of mathematics that deals with discrete elements that use distinct values. This field encompasses a wide array of topics like graph theory, logic, counting, and summation notation, which are fundamental to computer science and various branches of mathematics. In the context of our textbook problem, discrete mathematics principles are applied through the use of summation notation and the Kronecker's delta.Understanding discrete mathematics is essential when dealing with structures that are fundamentally discrete rather than continuous. As numbers in a sum are distinct and separate, these summations fall under this domain. When approaching problems in discrete mathematics, it's important to think iteratively and understand that every element can often be counted or listed. Such practices are at the heart of the problem-solving strategies used when dealing with nested summations and other similar constructs.

Summation Notation

Summation notation, commonly represented by the Greek letter sigma (\( \Sigma \)), is a convenient way to express the addition of a sequence of numbers. It is a shorthand that saves us from writing long, cumbersome sums and is essential in both simple and complex mathematical computations. The general notation is \[ \sum_{i=m}^{n} a_i \], where 'm' is the lower bound, 'n' is the upper bound, and \( a_i \) is the expression to be summed over the index i.

The detailed interpretation of summation notation involves understanding that the 'i' is an index that varies from 'm' to 'n', and for each value of 'i', the expression \( a_i \) is computed and added to the total sum. In the expression from our textbook example, \( j+3 \) represents the terms to be added, while 'i' sets the limit for the variable 'j'. Learning how to decode and compute summation notation is fundamental for anyone studying mathematics or related fields where quantitative analysis is required.

The value of correct summation notation cannot be overstated. It is a powerful tool for simplifying and managing calculations within a wide range of mathematical problems, from the simplest to the most complex, including the nested summations as illustrated in the textbook exercise.