Problem 5 - A Gentle Introduction to the Art of Mathematics

Question

Show by counterexample that ' 9 (divisibility) is not symmetric as a relation on \(\mathbb{Z}\).

Step-by-Step Solution

Answer

The relation 'divisibility by 9' is not symmetric because 9 divides 81, but 81 does not divide 9.

Step 1: Understand the Symmetric Property

A relation R on a set is symmetric if for every a and b in the set, whenever a is related to b (aRb), then b is related to a (bRa). In the case of divisibility, this means if a divides b, then b must also divide a for the relation to be symmetric.

Step 2: Define the Relation

The relation given is 'divisibility by 9'. This means for any integers a and b, if a divides b (denoted as a|b), then b should also divide a if the relation is symmetric.

Step 3: Choose Elements

Select specific integers to test this property. Let a = 9 and b = 81. Clearly, 9 divides 81 because 81 = 9 * 9.

Step 4: Test Symmetry

Now check if the reverse is true, i.e., whether 81 divides 9. For 81 to divide 9, there must exist an integer k such that 9 = 81 * k. This is not true because 9 is not a multiple of 81.

Step 5: Conclude the Counterexample

From the previous step, since 81 does not divide 9, the relation 'divisibility by 9' is not symmetric as it fails the symmetry requirement for these chosen numbers.

Key Concepts

relation properties

In mathematics, a relation on a set is a connection between pairs of elements in the set. To check specific attributes of these relations, we often evaluate their properties. One significant property is symmetry. A relation R on a set is symmetric if, for any two elements a and b in the set, whenever a is related to b, then b is also related to a. Symmetry means the relationship works both ways. For example, equality is symmetric because if a = b, then b = a.

Other important properties include:

• Reflexivity: Every element is related to itself. Formally, for all a in the set, aRa.
• Transitivity: If a is related to b, and b is related to c, then a is related to c. Formally, if aRb and bRc, then aRc.

Understanding these properties helps identify and classify different types of relations, offering valuable insight into their mathematical structures.

counterexample in mathematics

A counterexample is a specific case that disproves a general statement or proposition. In mathematics, finding a counterexample often involves identifying an instance where the proposed rule fails. This technique is powerful for testing hypotheses and validating theorems. For instance, the exercise demonstrates a counterexample to show that divisibility by 9 is not symmetric. The integers chosen, 9 and 81, illustrate that while 9 divides 81, the reverse is not true - 81 does not divide 9.

To effectively construct a counterexample:

• Identify the property you wish to invalidate.
• Choose specific instances or elements from the set.
• Verify you’ve found a case where the property fails.
  
  Counterexamples are crucial in mathematics because they help refine and correct assumptions, ensuring theories are robust and accurate.

integer divisibility

Divisibility is a fundamental concept in number theory. An integer a is said to divide another integer b if there exists an integer k such that b = a * k. This concept is denoted as a|b. For example, 3 divides 9 because 9 = 3 * 3.

Important aspects of divisibility include:

• Factors and Multiples: If a divides b, then a is a factor of b, and b is a multiple of a.
• Prime Numbers: A prime number is only divisible by 1 and itself.
• Greatest Common Divisor (GCD): The largest integer that divides two numbers without leaving a remainder.

In the context of the exercise, divisibility is used to explore the symmetry property. By examining specific integers, such as 9 and 81, it is shown that while 9 divides 81, 81 does not divide 9. This lack of mutual divisibility illustrates that the relation 'divisibility by 9' is not symmetric.