MAT2611 ASSIGNMENT 1 2023
Problem 1
Exercise 2.14
1. 𝐴 = {𝑥 ∈ ℚ ∶ 𝑥 2 = 2}
The set 𝐴 contains all the rational numbers whose square is 2.
To solve the equation 𝑥 2 = 2:
𝑥2 = 2
𝑥2 − 2 = 0
(𝑥 + √2)(𝑥 − √2) = 0
𝑥 = −√2 or 𝑥 = √2
The equation has two solutions which are irrational.
There is no rational number whose square is 2.
The set 𝐴 has no elements at all, so it is the empty set.
𝐴=∅
2. 𝐵 = {𝑥 ∈ ℝ ∶ 𝑥 2 + 1 = 0}
𝐵 = {𝑥 ∈ ℝ ∶ 𝑥 2 = −1}
The set 𝐵 contains all real numbers whose square is equal to −1.
Now, the square of any real number is positive.
That is for any 𝑥 ∈ ℝ, 𝑥 2 ≥ 0
The equation 𝑥 2 + 1 = 0 has no real solutions.
The set 𝐵 is empty.
𝐵=∅
Problem 1
Exercise 2.14
1. 𝐴 = {𝑥 ∈ ℚ ∶ 𝑥 2 = 2}
The set 𝐴 contains all the rational numbers whose square is 2.
To solve the equation 𝑥 2 = 2:
𝑥2 = 2
𝑥2 − 2 = 0
(𝑥 + √2)(𝑥 − √2) = 0
𝑥 = −√2 or 𝑥 = √2
The equation has two solutions which are irrational.
There is no rational number whose square is 2.
The set 𝐴 has no elements at all, so it is the empty set.
𝐴=∅
2. 𝐵 = {𝑥 ∈ ℝ ∶ 𝑥 2 + 1 = 0}
𝐵 = {𝑥 ∈ ℝ ∶ 𝑥 2 = −1}
The set 𝐵 contains all real numbers whose square is equal to −1.
Now, the square of any real number is positive.
That is for any 𝑥 ∈ ℝ, 𝑥 2 ≥ 0
The equation 𝑥 2 + 1 = 0 has no real solutions.
The set 𝐵 is empty.
𝐵=∅
