Answer:
a. ∀ a, b ∈ Z, if a < 0 ∧ b < 0 ⇒ a + b < 0
b. ∀ a, b ∈ Z, if a > 0 ∧ b > 0 ⇒ a - b < 0 ∨ a - b > 0
c. ∀ a, b ∈ Z, a² + b² ≥ (a + b)
d. ∀ a, b ∈ Z, |ab| = |a||b|
Step-by-step explanation:
a. The sum of two negative integers is negative.
First, we take the quantifier ∀ and write the two integers a and b as a element of the set of integers Z as ∀ a, b ∈ Z. Now, if a and b are negative, we write if a < 0 ∧ b < 0 where the logical operator ∧ represents "and". So, we have the statements ∀ a, b ∈ Z, if a < 0 ∧ b < 0. Then finally, we use the logical operator ⇒ to imply the summation of the two integers as ⇒ a + b. And since we are to show that their sum is less than zero, we write ⇒ a + b < 0.
Combining this statement with the previous expressions, we have
∀ a, b ∈ Z, if a < 0 ∧ b < 0 ⇒ a + b < 0
b. The difference of two positive integers is not necessarily positive.
First, we take the quantifier ∀ and write the two integers a and b as a element of the set of integers Z as ∀ a, b ∈ Z. Now, if there exists a and b that are positive, we write if a > 0 ∧ b > 0 where the logical operator ∧ represents "and". So, we have the statements ∀ a, b ∈ Z, a > 0 ∧ b > 0. Then finally, we use the logical operator ⇒ to imply the difference of the two integers as ⇒ a - b. And since we are to show that their difference is either less than zero or greater than zero, we write ⇒ a - b < 0 ∨ a - b > 0 where ∨ implies the logical operator "or".
Combining this statement with the previous expressions, we have
∀ a, b ∈ Z, if a > 0 ∧ b > 0 ⇒ a - b < 0 ∨ a - b > 0
c. The sum of the squares of two integers is greater than or equal to the square of their sum.
First, we take the quantifier ∀ and write the two integers a and b as a element of the set of integers Z as ∀ a, b ∈ Z. Now, since we are dealing with the sum of the squares of each integer, we write a² + b². And, we are to show that this sum is greater than the sum of the individual integers, we write their sum as a + b, and we show the previous sum to be greater than or equal to this as a² + b² ≥ (a + b).
Combining this statement with the previous expressions, we have
∀ a, b ∈ Z, a² + b² ≥ (a + b)
d. The absolute value of the product of two integers is the product of their absolute values.
First, we take the quantifier ∀ and write the two integers a and b as a element of the set of integers Z as ∀ a, b ∈ Z. Now, since we are to show that absolute value of the product of the integers equals the product of the absolute value of the individual integers, we write the that absolute value of the product of the integers as |ab| and the product of the absolute value of the individual integers as |a||b|.
Since we are to show that this expression and the previous expression are equal, we write |ab| = |a||b|.
Combining this statement with the previous expressions, we have
∀ a, b ∈ Z, |ab| = |a||b|
