Question

Express each of these statements using predicates, quantifiers, logical connectives, and mathematical operators where the domain consists of all integers.

a) The product of two negative integers is positive.

b) The average of two positive integers is positive.

c) The difference of two negative integers is not necessarily negative.

d) The absolute value of the sum of two integers does not exceed the sum of the absolute values of these integers.

Short answer

Each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers can be expressed.

Step-by-step solution

Determining the product of two negative integers is positive

a)

Here, $\mathrm{xy}>0$represents the product of two integers which is greater than zero.

$(\mathrm{x}<0)\wedge (\mathrm{y}<0)$represents x and y which is less than zero.

$\forall \mathrm{x}\forall \mathrm{y}\left(\right(\mathrm{x}<0)\wedge (\mathrm{y}<0\left)\right)\to (\mathrm{xy}>0))$ denotes when x and y are less than zero, So the product of two is positive.

Determining the average of two positive integers is positive

b)

Here, $(\mathrm{x}>0)\wedge (\mathrm{y}>0)$represents when x and y are greater than zero. The average of two number is greater than zero denoted byrole="math" localid="1668585724544" $(\frac{\mathrm{x}+\mathrm{y}}{2}>0)$ . When x and y are greater than zero then the two average is greater than zero can be denoted by$\forall \mathrm{x}\forall \mathrm{y}\left(\right(\mathrm{x}>0)\wedge (\mathrm{y}>0)\to \left(\frac{\mathrm{x}+\mathrm{y}}{2}>0\right)$

Determining the difference of two negative integers is not necessarily negative

c)

Here, $(\mathrm{x}<0)\wedge (\mathrm{y}<0)$represents x and y are less than zero. $\mathrm{x}-\mathrm{y}\ge 0$represents difference of two number when greater than zero. $\exists \mathrm{x}\exists \mathrm{y}\left(\right(\mathrm{x}<0)\wedge (\mathrm{y}<0)\wedge (\mathrm{x}-\mathrm{y})\ge 0)$ denotes when x and y are less than zero, So the difference of two when greater than zero.

Determining the absolute value of the sum of two integers does not exceed the sum of the absolute values of these integers

d)

The absolute value of sum of two integers are $|\mathrm{x}+\mathrm{y}|\forall \mathrm{x}\forall \mathrm{y}\left(\right|\mathrm{x}+\mathrm{y}|=|\mathrm{x}|+|\mathrm{y}\left|\right)$denotes x and y for the absolute value of the sum of two integers should be not exceed to the absolute values sum of these integers.