TRUE OR FALSE ? Justify your answer.
a. There exist real numbers a and b such (a+b)^2 = a^2 + b^2.
b. For any real number x, there exists a unique number -x such that x + (-x) = 0.
c. Any integer n is either odd or even.
d. Any real number x can be written as a fraction.
Write the statements from above using the logical quantifiers ∀ and ∃ .
Negate the statements from above (first question) using the logical quantifiers ∀ and ∃ .
(a) true: a=0 or b=0
∃ a,b ∊ R : (a+b)^2 = a^2+b^2
(b) true
(c) true
(d) false
a. FALSE. To check if the statement is true or false, we need to justify our answer by providing a counterexample. Let's assume a = 1 and b = -1. Then, (1+(-1))^2 = (0)^2 = 0, but a^2 + b^2 = 1^2 + (-1)^2 = 1 + 1 = 2. Since 0 is not equal to 2, the given statement is false.
b. TRUE. To justify this, we can use the concept of additive inverse. For any real number x, there exists a unique number -x such that x + (-x) = 0. So, for every x, there is a unique -x that satisfies this condition, making the statement true.
c. TRUE. This statement can be written using the logical quantifiers as ∀n ∈ Z (n is odd ∨ n is even). In other words, for any integer n, it is either odd or even. This statement is true as every integer can be divided into two categories: odd or even.
d. FALSE. The statement "Any real number x can be written as a fraction" is false. This can be justified by providing a counterexample. Let's consider the real number π (pi). π is an irrational number, which means it cannot be expressed as a fraction. Therefore, the given statement is false.
The negation of the statements using the logical quantifiers are:
a. ¬∃a, b ∈ ℝ ((a + b)^2 = a^2 + b^2)
b. ¬∀x ∈ ℝ ∃y (x + y = 0)
c. ¬∀n ∈ ℤ (n is odd ∨ n is even)
d. ¬∀x ∈ ℝ ∃m, n (x = m/n)