Q52. - Algebra 2

Question

Which of the following properties hold for inequalities? Explain your reasoning or give a counterexample.

1. Reflexive     b.   Symmetric        c. Transitive

Step-by-Step Solution

Answer

1. The reflexive property does not hold for inequalities.

        The counterexample for the inequality is: $\mathbf{1}\mathbf{>}\mathbf{1}$.

2. The symmetric property does not hold for inequalities.

   The counterexample for the inequality is: $\mathbf{2}\mathbf{>}\mathbf{1}$.

3. The transitive property holds for inequalities because if the given inequality is $\mathbf{a}\mathbf{>}\mathbf{b}$ then for numbers $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$, it can be noticed that if $\mathbf{a}$ is greater than $\mathbf{b}$ and $\mathbf{b}$ is greater than $\mathbf{c}$, then $\mathbf{a}$ is necessarily greater than $\mathbf{c}$.

a. Step 1 ­- Description of step.

The reflexive property states that $a=a$.

Step 2 ­- Description of step.

Consider the inequality as $a>b$.

For reflexive property $a>a$.

As it can be noticed that for any number $a$, it cannot be possible that $a>a$.

Therefore, the inequality does not satisfy the reflexive property.

The counterexample for the inequality is: $1>1$.

As it can be noticed that the number 1 cannot be greater than itself.

Therefore, the reflexive property is not satisfied.

Step 3 ­- Write the conclusion.

The reflexive property does not hold for inequalities.

The counterexample for the inequality is: $1>1$.

b. Step 1 ­- Description of step.

The symmetric property states that if $a=b$ then $b=a$.

Step 2 ­- Description of step.

Consider the inequality as $a>b$.

For symmetric property $b>a$.

As it can be noticed that for numbers $a$ and $b$, it cannot be possible that simultaneously $a>b$ and $b>a$.

Therefore, the inequality does not satisfy the symmetric property.

The counterexample for the inequality is: $2>1$.

As it can be noticed that the number 2 is greater than 1 but 1 cannot be greater than 2.

Therefore, the symmetric property is not satisfied.

Step 3 ­- Write the conclusion.

The symmetric property does not hold for inequalities.

The counterexample for the inequality is: $2>1$.

c. Step 1 ­- Description of step.

The transitive property states that if $a=b$, $b=c$ then $a=c$.

Step 2 ­- Description of step.

Consider the inequality as $a>b$ and $b>c$.

For transitive property $a>c$.

As it can be noticed that for numbers $a$, $b$ and $c$, it can be noticed that if $a$is greater than $b$ and $b$ is greater than $c$, then $a$ is necessarily greater than $c$.

Therefore, the inequality satisfies the transitive property.

Step 3 ­- Write the conclusion.

The transitive property holds for inequalities.