Problem 52 - Discrete Mathematics with Applications

Question

Expand each. $$\sum_{1 \leq i \leq j<3}\left(a_{i}+a_{j}\right)$$

Step-by-Step Solution

Answer

The short answer for the given summation \(\sum_{1 \leq i \leq j<3}\left(a_{i}+a_{j}\right)\) is: \(\sum_{1 \leq i \leq j<3}\left(a_{i}+a_{j}\right) = 3a_1 + 3a_2\)

Step 1: Identify the Allowed Values for i and j

The given expression specifies the summation with the indices i and j, where \(1 \leq i \leq j < 3\). The allowed values of i are: 1 and 2. For each i, we can find the allowed values for j.

Step 2: Calculate the Sums for All Pairs of Indices

For each possible pair of indices (i, j) which meet the given inequality, calculate the sum of the corresponding elements: - For i = 1: - j = 1: \(a_1 + a_1\) - j = 2: \(a_1 + a_2\) - For i = 2: - j = 2: \(a_2 + a_2\) Notice that there are no other values of j that satisfy the inequality and the summation's definition.

Step 3: Add The Summands Found in Step 2

Now that we have found the terms to be included in the summation, let's find the sum: \(\sum_{1 \leq i \leq j<3}\left(a_{i}+a_{j}\right) = (a_1 + a_1) + (a_1 + a_2) + (a_2 + a_2)\)

Step 4: Simplify the Expression

Finally, we can simplify the expression: \((a_1 + a_1) + (a_1 + a_2) + (a_2 + a_2) = 2a_1 + a_1 + a_2 + a_2 + a_2 = 3a_1 + 3a_2\) So, the expanded expression for the given summation is: \[\sum_{1 \leq i \leq j<3}\left(a_{i}+a_{j}\right) = 3a_1 + 3a_2\]

Key Concepts

Discrete Mathematics

Discrete mathematics is a branch of mathematics focusing on the study of mathematical structures that are fundamentally discrete rather than continuous. In practice, this means that we deal with objects that can only take on distinct, separated values, like integers.

Within the realm of discrete mathematics, we encounter concepts like sets, graphs, and algorithms, as well as different types of notation, including sigma notation, which is used to represent summation. Problems in disciplines such as computer science, information theory, and combinatorics can often be modeled and solved using discrete mathematics. For example, in understanding algorithms, one needs to be familiar with sequences and sums—a perfect segue into sigma notation.

Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of numbers. This notation uses the Greek letter sigma (\f\(\f\bigsum\f\)) as a symbol to indicate summation. A typical sigma notation includes an expression for the terms to be summed, an index that keeps track of the terms, and limits that specify the starting and ending values of the index.

For instance, considering our exercise \f\(\f\bigsum_{1 \f\bigsum i \f\bigsum j<3}\f(af_{i}af_{j}\f)\f\), it's written using sigma notation and describes the sum of all the terms \f(af_{i}af_{j}\f) where \f\(i\f\) and \f\(j\f\) satisfy certain conditions. The process of 'expanding' sigma notation means writing out all the individual terms specified by the notation, rather than the compact sigma representation.

Mathematical Induction

Mathematical induction is a proof technique that is used primarily to prove that a statement is true for all natural numbers. It consists of two crucial steps: the base case and the inductive step. In the base case, the statement is proven to be true for the first natural number. Next, in the inductive step, one assumes the statement is true for some natural number \f\(k\f\), and then demonstrates that it must also be true for \f\(k + 1\f\).

This technique is akin to falling dominos; once you show the first one falls (the base case), and that any domino will knock over the next one (the inductive step), you can conclude that all the dominos will fall. Mathematical induction is central to many proofs and arguments in discrete mathematics, but is distinct from the practice of iterating through the elements of a sum as done in the sigma notation example from our exercise.