Problem 49 - Algebra 1: Concepts and Skills

Question

Determine whether the statement is true or false. If it is false, give a counterexample. If \(a>b,\) then \(a \cdot 0\) is greater than \(b \cdot 0\)

Step-by-Step Solution

Answer

The statement is false. A counterexample is when \(a=7\) and \(b=2\), then both \(a \cdot 0 = 0\) and \(b \cdot 0 = 0\), thus one is not greater than the other.

Step 1: Understand the given statement

The given statement is: If \(a>b,\) then \(a \cdot 0\) is greater than \(b \cdot 0\). This means that the product of \(a\) and zero is said to be greater than the product of \(b\) and zero.

Step 2: Analysing the Concept

We know that when a number is multiplied by zero, the result is zero. This is an established mathematical rule or concept. Knowing this, we can predict the result of \(a \cdot 0\) and \(b \cdot 0\). Both are zero, regardless of the values of \(a\) and \(b\).

Step 3: Verifying the Statement

We now apply this property to the given statement. Regardless of whether \(a > b\), since both \(a \cdot 0\) and \(b \cdot 0\) are equal to zero, one cannot be said to be greater than the other. Hence, the statement is false.

Step 4: Provide a Counterexample

Choose any \(a > b\). For instance, let \(a= 7\) and \(b= 2\). Then, according to the given statement, \(7 \cdot 0\) which is 0, should be greater than \(2 \cdot 0\), which is also 0. However, we know that this is not true as both are equal.

Key Concepts

Zero Product Property

The zero product property is a fundamental principle in algebra that states if you multiply any number by zero, the result is always zero. Mathematically, this is expressed as if \( a \) is any real number, then \( a \cdot 0 = 0 \).

In the context of the exercise provided, this principle is crucial. When the statement suggests \( a \cdot 0 \) is greater than \( b \cdot 0 \), it misuses the zero product property because both expressions will invariably evaluate to zero, making \( a \cdot 0 \) neither greater than nor less than \( b \cdot 0 \). To avoid falling into such traps, it's essential to remember that zero acts as a 'nullifier' in multiplication, rendering any comparison of values irrelevant.

Inequalities

Inequalities are statements about the relative size or order of two objects, expressed using signs such as \( > \), \( < \), \( \geq \), and \( \leq \). They indicate that one quantity is greater than, less than, greater than or equal to, or less than or equal to another quantity. However, when zero is introduced into an inequality through multiplication, as in our exercise, the typical rules of inequalities no longer apply.

This is because, as previously mentioned, any number multiplied by zero results in zero, and zero is not greater or less than itself; it is equal to itself. Hence, no matter what values \( a \) and \( b \) represent in the inequality \( a \cdot 0 > b \cdot 0 \), the inequality falls apart.

Counterexamples

Counterexamples are a powerful tool in mathematics used to disprove statements or assertions. A counterexample is a specific case for which the general statement does not hold true. It can prove that a statement is false with just a single instance.

When we say \( a \cdot 0 > b \cdot 0 \), producing a counterexample is simple because any values we choose for \( a \) and \( b \), maintaining the condition \( a > b \), we always obtain zero when multiplying by zero. Thus, the original statement is always false, no matter the selected values for \( a \) and \( b \), making it an excellent demonstration of the concept of counterexamples.