Which of the following set is empty set? D = x : x ∈ N , x 2 + 4 = 0

D = x : x ∈ N , x 2 + 4 = 0 We know that, a set with no elements in it is called an empty set. It is denoted by { } or ϕ . Natural numbers = 1 , 2 , 3 , 4 , 5 , 6 , … Here, x 2 + 4 = 0 Discriminant b 2 - 4 a c = 0 - 4 1 4 = - 16 < 0 No real solutions because the discriminant is negative ∴ No elements in the set B . Therefore, it is a null set.

Which of the following set is empty set? F = { x : x ∈ W , x + 3 < 3 }

Given: F = { x : x ∈ W , x + 3 < 3 } We know that, a set with no elements in it is called an empty set. It is denoted by { } or ϕ . Whole numbers = 0 , 1 , 2 , 3 , … If we take x = 0 then x + 3 = 0 + 3 = 3 If we take x = 1 then x + 3 = 1 + 3 = 4 > 3 So, 0 is the element of set E because it satisfies the given equation. Therefore, it is not a null set.

Which of the following set is empty set? G = { x : x ∈ Q , 1 < x < 2 }

Given: G = { x : x ∈ Q , 1 < x < 2 } i.e., x is a rational number. We know that, If a and b are two rational numbers, then a + b 2 is a rational number between a and b such that a < a + b 2 < b . So, the rational number between 1 and 2 is 1 + 2 2 = 3 2 Here, the set is not empty. Therefore, it is not a null set.

Which of the following are pair of equal sets? C = x : x ∈ Z , x 2 ≤ 4 and D = x : x ∈ Z , x 2 - 4 = 0

Given: C = x : x ∈ Z , x 2 ≤ 4 and D = x : x ∈ Z , x 2 - 4 = 0 We know that, two sets A and B are said to be equal sets if they have exactly the same elements and it is written as A = B . Here, x ∈ Z and x 2 ≤ 4 If x = - 2 , then x 2 = ( - 2 ) 2 = 4 = 4 . If x = - 1 , then x 2 = ( - 1 ) 2 = 1 < 4 . If x = 0 , then x 2 = ( 0 ) 2 = 0 < 4 . If x = 1 , then x 2 = ( 1 ) 2 = 1 < 4 . If x = 2 , then x 2 = ( 2 ) 2 = 4 = 4 . So, C = { - 2 , - 1 , 0 , 1 , 2 } Now, D = x : x ∈ Z , x 2 = 4 Here, x ∈ Z and x 2 = 4 . If x = - 2 then x 2 = ( - 2 ) 2 = 4 = 4 . If x = 2 , then x 2 = ( 2 ) 2 = 4 = 4 . So, D = { - 2 , 2 } . ∴ C ≠ D because the elements in the both the sets are not equal.

Which of the following are pair of equal sets? E = { x : x ∈ Z , x - 3 = 0 } and F = x : x ∈ Z , x 2 - 9 = 0

Given: E = { x : x ∈ Z , x - 3 = 0 } and F = x : x ∈ Z , x 2 - 9 = 0 We know that, two sets A and B are said to be equal sets if they have exactly the same elements, and it is written as A = B . Here, x ∈ Z and x - 3 = 0 On solving the equation, we get x = 3 So, E = { 3 } Now, F = x : x ∈ Z , x 2 - 9 = 0 Here, x ∈ Z and x 2 - 9 = 0 . On solving the equation, we get x 2 = 9 ⇒ x = ± 3 So, F = { - 3 , 3 } . ∴ E ≠ F because the elements in the both the sets are not equal.