Question

Express each of these mathematical statements using predicates, quantifiers, logical connectives, and mathematical operators.

Answer 1

Answer:

a. a × b > 0 ∀ a,b ∈ R : a,b < 0

b. a - a = 0 ∀ a ∈ R

Step-by-step explanation:

a. Let a and b be the numbers. Since it says product of two numbers is greater than zero, we write a × b > 0. Since a and b are real numbers, we write a,b ∈ R where ∈ denotes element of a set and R is the set of real numbers. We then use the connective ∀ which denotes "for all" to join a × b > 0 with a,b ∈ R. So, we write a × b > 0 ∀ a,b ∈ R. Since a and b are negative, we write a,b < 0. We now use the connective : which denotes "such that" to combine a × b > 0 ∀ a,b ∈ R with a,b < 0 to give a × b > 0 ∀ a,b ∈ R : a,b < 0. So, the expression is

a × b > 0 ∀ a,b ∈ R : a,b < 0

b. Let a be the number. Since we are looking for a difference, we write a - a. Since it is equal to zero, we write a - a = 0. Since a is an element of real numbers,R, we write a ∈ R, where ∈ denotes "element of". So, a ∈ R denotes a is an element of real numbers R. We combine these two expressions with the connective ∀ which denotes "for all" to give a - a = 0 ∀ a ∈ R. So, the expression is

a - a = 0 ∀ a ∈ R