Which one of the following is the empty set?a){x : x is a real number ...
The empty set is a set that does not contain any elements. In other words, it is a set with no members.
To determine which of the given options represents the empty set, we need to analyze the conditions specified in each option and check if there are any real numbers that satisfy those conditions.
Option a: {x : x is a real number and x^2 - 1 = 0}
- This option represents the set of real numbers that satisfy the equation x^2 - 1 = 0.
- Solving this equation, we get x = ±1.
- So, this set contains two elements: {1, -1}. Therefore, it is not the empty set.
Option b: {x : x is a real number and x^2 + 1 = 0}
- This option represents the set of real numbers that satisfy the equation x^2 + 1 = 0.
- However, this equation has no real solutions since the square of any real number is always non-negative.
- Therefore, there are no real numbers that satisfy this equation, and the set is empty.
Option c: {x : x is a real number and x^2 - 9 = 0}
- This option represents the set of real numbers that satisfy the equation x^2 - 9 = 0.
- Solving this equation, we get x = ±3.
- So, this set contains two elements: {3, -3}. Therefore, it is not the empty set.
Option d: {x : x is a real number and x^2 = x + 2}
- This option represents the set of real numbers that satisfy the equation x^2 = x + 2.
- Solving this equation, we get x = -1, x = 2.
- So, this set contains two elements: {-1, 2}. Therefore, it is not the empty set.
Therefore, option b is the correct answer as it represents the empty set.
Which one of the following is the empty set?a){x : x is a real number ...
Since, x2 + 1 = 0, given x2 = -1
x = ± i
Therefore, x is not real but x is real (given)
Hence, No value of x is possible.