Problem 20 - Discrete Mathematics with Applications

Question

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$7 \odot 7$$

Step-by-Step Solution

Answer

The operations for 7.7 in the given boolean algebra are as follows: - 7 + 7 (lcm): \(7 + 7 = 7\) - 7 . 7 (gcd): \(7 . 7 = 7\) - 7' (prime): \(7' = 10\)

Step 1: Compute 7 lcm 7 (7 + 7)

To compute the lowest common multiple (lcm) of 7 and 7, we look for the smallest number that is divisible by both 7 and 7. In this case, the lcm(7, 7) is 7 itself. So, 7 + 7 = 7.

Step 2: Compute 7 gcd 7 (7 . 7)

To compute the greatest common divisor (gcd) of 7 and 7, we look for the largest number that divides both 7 and 7. In this case, the gcd(7, 7) is 7 itself. So, 7 . 7 = 7.

Step 3: Compute 7 prime (7')

To compute the prime (') operation, we use the given formula x' = 70/x. In this case, x is 7, so 7' = 70/7. Therefore, 7' = 10.

Key Concepts

LCM (Least Common Multiple)

Understanding the concept of the Least Common Multiple (LCM) is essential for various areas of mathematics, including algebra and number theory. The LCM of two or more numbers is the smallest non-zero number that is a multiple of all the numbers in the set. For example, to find the LCM of 12 and 15, we list the multiples of each:

• Multiples of 12: 12, 24, 36, 48, 60, ...
• Multiples of 15: 15, 30, 45, 60, ...

The smallest number that appears on both lists is 60, hence the LCM of 12 and 15 is 60. This concept plays a crucial role in operations involving fractions and finding common denominators for addition and subtraction. It is also used in boolean algebra, as demonstrated in the example from the textbook where the LCM of 7 and 7 is straightforwardly 7.

GCD (Greatest Common Divisor)

The Greatest Common Divisor (GCD), also known as the greatest common factor or highest common factor, is the largest positive integer that divides two or more integers without leaving a remainder. Finding the GCD is a fundamental operation in elementary number theory.

Finding the GCD

To determine the GCD of two numbers, we consider all the factors of each number and then find the largest factor they have in common. For instance, the factors of 8 are 1, 2, 4, and 8, while the factors of 12 are 1, 2, 3, 4, 6, and 12. The common factors are 1, 2, and 4, with 4 being the greatest. Therefore, the GCD of 8 and 12 is 4. In the context of discrete mathematics and boolean algebra, the GCD operation helps simplify expressions and is symbolized by a dot, as seen in the solved exercise, where the GCD of 7 and 7 is simply 7.

Positive Factors

Positive factors of a number are whole numbers that divide the original number exactly, without leaving a remainder. These are critical in various branches of mathematics, particularly in discrete mathematics and number theory. For instance, the positive factors of 70 include 1, 2, 5, 7, 10, 14, 35, and 70. Each pair of factors, when multiplied, will result in the original number (e.g., 2 × 35 = 70).

Factor Pairs

Each factor corresponds to a 'partner' that multiplies with it to give the original number. Identifying these is useful in solving equations, factoring polynomials, and simplifying radicals. Furthermore, knowledge of positive factors is vital in advanced areas such as cryptography.

Discrete Mathematics

Discrete mathematics is a branch of mathematics that deals with discrete, usually countable, sets as opposed to continuous sets. It encompasses several fields such as set theory, graph theory, and combinatorics, and is the foundation of computer science and cryptography.

In discrete mathematics, boolean algebra is a sub-field that represents one of the simplest forms of logic that also has applications in circuit design and digital computing. It is centered around variables that can have only two possible values, typically denoted as 0 and 1, and features operations such as AND, OR, and NOT. In the context of the exercise, boolean algebra is defined over a set of positive divisors of 70 under specific operations, emphasizing its significant role in establishing logical structures and computational frameworks.