If G₁ = {x : x ∈ ℚ and x² = 2}, what type of set is G₁?
Answer and explanation
Correct answer: Empty set
Over the real numbers, x² = 2 has the solutions x = √2 and x = −√2. However, √2 is irrational, and its negative is also irrational. Because the definition restricts x to ℚ, neither real solution is allowed. Thus G₁ has no rational element and is the empty set, G₁ = ∅. The two real solutions do not make it a two-element rational set.
Frequently asked questions
What is the correct answer to this question?
Empty set
Why is this the correct answer?
Over the real numbers, x² = 2 has the solutions x = √2 and x = −√2. However, √2 is irrational, and its negative is also irrational. Because the definition restricts x to ℚ, neither real solution is allowed. Thus G₁ has no rational element and is the empty set, G₁ = ∅. The two real solutions do not make it a two-element rational set.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: The Empty Set, Finite and Infinite Sets, Equal Sets.
