Which of the following set is the empty set ?

  1. {x: x + 4 = 7}
  2. {x : 7 < x < 8, x is a natural number}
  3. {x:  x2 - x = 2}
  4. {x: x2 - 3x + 2 = 0} 

Answer (Detailed Solution Below)

Option 2 : {x : 7 < x < 8, x is a natural number}
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Detailed Solution

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Concept:

The empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

 

Calculation:

1. x: x + 4 = 7

⇒ x = 7 - 4 = 3

∴ {x: x + 4 = 7} = {2} 

It is not a empty set.

 

2. {x : 7 < x < 8, x is a natural number} will be a null set because there is NO natural number between numbers 7 and 8.

 

3.  x2​ - x  = 2

⇒ x(x - 1) = 2

⇒ x = 2 OR x = -1

∴  It is also not a empty set.

 

4. x2 - 3x + 2 = 0

⇒ x2 - 2x - x + 2 = 0

⇒ x(x - 2) - 1 (x - 2) = 0

⇒ (x- 2)(x - 1) = 0

⇒ x = 2 OR x = 1

∴  It is also not a empty set.

 

Hence, option (2) is correct.

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