Exercise: Properties of Inequality

This demonstrates the Multiplicative Property of Inequality: when both sides of an inequality are multiplied by the same number, the direction of the inequality is:

• preserved if the number is positive
• reversed if the number is negative

For example, if $a > b$ and $c > 0$, then $a \times c > b \times c$.

Formal: $\text{For all } a, b, c \in \mathbb{R},\ a > b \land c > 0 \Rightarrow a \times c > b \times c$

This is a generalized case of the Additive Property of Inequality: if $a < b$ and $c < d$, then their sums also follow the inequality.

That is, $a + c < b + d$.

The formal statement: $a < b \land c < d \Rightarrow a + c < b + d$

This demonstrates the Multiplicative Property of Inequality: when both sides of an inequality are multiplied by the same number, the direction of the inequality is:

• preserved if the number is positive
• reversed if the number is negative

For example, if $a < b$ and $c < 0$, then $a \times c > b \times c$.

Formal: $\text{For all } a, b, c \in \mathbb{R},\ a < b \land c < 0 \Rightarrow a \times c > b \times c$

This demonstrates the Multiplicative Property of Inequality: when both sides of an inequality are multiplied by the same number, the direction of the inequality is:

• preserved if the number is positive
• reversed if the number is negative

For example, if $a < b$ and $c < 0$, then $a \times c > b \times c$.

Formal: $\text{For all } a, b, c \in \mathbb{R},\ a < b \land c < 0 \Rightarrow a \times c > b \times c$

This demonstrates the Additive Property of Inequality: if one number is less than another, adding the same quantity to both sides preserves the inequality.

That is, if $a < b$, then $a + c < b + c$.

The formal statement: $\text{For all } a, b, c \in \mathbb{R},\ a < b \Rightarrow a + c < b + c$

This illustrates the Trichotomy Property: for any real number a and any constant, exactly one of the following is true:

• $a = b$
• $a > b$
• $a < b$

In this case, \dfrac{1}{15} is being compared to b.

Formal definition: $\forall a, b \in \mathbb{R},\ a = b \text{ or } a > b \text{ or } a < b$

This demonstrates the Transitive Property of Inequality: if one number is less than a second, and the second is less than a third, then the first is less than the third.

That is, if $a < b \land b < c$, then $a < c$.

The formal statement is: $\text{For all } a, b, c \in \mathbb{R},\ a < b \land b < c \Rightarrow a < c$.

This demonstrates the Transitive Property of Inequality: if one number is greater than a second, and the second is greater than a third, then the first is greater than the third.

That is, if $a > b \land b > c$, then $a > c$.

The formal statement is: $\text{For all } a, b, c \in \mathbb{R},\ a > b \land b > c \Rightarrow a > c$.

This demonstrates the Transitive Property of Inequality: if one number is greater than a second, and the second is greater than a third, then the first is greater than the third.

That is, if $a > b \land b > c$, then $a > c$.

The formal statement is: $\text{For all } a, b, c \in \mathbb{R},\ a > b \land b > c \Rightarrow a > c$.

This is a generalized case of the Additive Property of Inequality: if $a > b$ and $c > d$, then their sums also follow the inequality.

That is, $a + c > b + d$.

The formal statement: $a > b \land c > d \Rightarrow a + c > b + d$