1) Which of the following collections are sets?
    i) The collection of all prime numbers between 7 and 19.
   ii) The collection of rich persons in India.

         i) A natural number ‘p’ is called a prime number if it is greater than one and its only factors are ‘1’ and ‘p’.
            Prime numbers between 7 and 19 are 11, 13, 17.
            ∴ Given collection is a set and contains 11, 13, 17.
        ii) The concept of ‘rich person’ is vague and there is no  rule for deciding whether a particular person is rich or not.
            ∴ Given collection is not a set.

2) Which of the following collections are set?
    i) Collection of all factors of 50 which are greater than 6.
   ii) The collection of all integer which are not natural numbers.

      i) The factor of 50 are 1, 2, 5, 10, 25 and 50 and out of these, 10, 25 and 50 are greater than 6.
         ∴ Given collection is a set and contains 10, 25 and 50.
     ii) Integers are 0, ±1, ±2, ±3…. And natural numbers are 1, 2, 3,…..
         ∴ Integers which are not natural numbers are 0, -1, -2, -3,…..
         ∴ Given collection is a set and contains 0, -1, -2, -3,…..

3) Write the following sets by roster method:
     i) The set of all natural number ‘x’ such 4x + 9 < 50
    ii) The set of all integer ‘x’ such that x² + 5x + 6 = 0

         i) 4x + 9 < 50 ⇒  4x < 41⇒ x  < 41/4 ⇒ x < 10.25
            Since x is a natural number, so x can take values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
            ∴ Given set = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

         ii) 
             -3, -2 are both integers .
             Given set = {-3, -2}.

4) Write the following set  by roaster method: The set of all nature number ‘x’ such that |x| ≤ 4 

      Since, x is positive, we have |x| = x
      ∴ |x| ≤ 4 implies x ≤ 4
      ∴ The possible values of x are 1, 2, 3, 4.
      ∴ Given set = {1, 2, 3, 4}.

5) Describe the following set by set property method:
    {6, 10, 14, 18}
     
     The given set is {6, 10, 14, 18}
     We observe that 6 = 4 (1) + 2,
                             10 = 4(2) + 2,
                             14 = 4(3) + 2
                             18 = 4(4) + 2
     ∴ The given set is equal to
     {x : x = 4n + 2, n ≤ 4, n  N}.

6) Describe the following set by set property method:
    {2, 3, 5, 7, 11}
       
      The given set is {2, 3, 5, 7, 11}.
      We observe that 2, 3, 5, 7, 11 are all prime numbers.
      ∴ Given set = {x:x is a prime number, x ≤ 11}.

7) Describe the following set  by set property method:
   {1/4, 1/8. 1/16, 1/32, 1/64}
     
      The given set is {1/4, 1/8. 1/16, 1/32, 1/64}
      We observe that 1/4= 1/2², 1/8 = 1/2³, 1/16 = 1/2⁴, 1/32 = 1/2⁵, 1/64 = 1/2⁶
      ∴ The elements of the given set are of the for, 1/2ⁿ, where n = 2, 3, 4, 5, 6.
      ∴ Given set = {x:x = 1/2ⁿ,  n N, 2 ≤ n ≤ 6}.

8) Describe the following sets by roster method:
    i) {x : x² + 5x + 6 = 0, x  N}
   ii) {x : x² + 6x + 8 = 0, x  Z}
        i) The equation is x² + 5x + 6 = 0
            ∴ x = -2, -3
            But -2, -3 are not in N.       ∴ Given set = {}
       ii) x² + 6x + 8 = 0 ⇒  x = -2, -4
           -2, -4 are in Z.                   ∴ Given set = {-2, -4}.

9) Describe the following sets by roster method:
     i) {x : 4x + 7 < 25, x  N}
    ii) the set of all letters in the word.
        TRIGONOMETRY.
        
          i) 4x + 7      
         ii) Different letters in the word  TRIGONOMETRY are T, R, I, G, O, N, M, E, Y.
             ∴ Given set = {T, R, I, G, O, N, M, E, Y}.

10) Which of the following sets are null sets?
      i) The set A of all prime numbers lying between 15 and 19.
     ii) A = {x : x is a natural number between 1 and  2 }
           
        i) Natural numbers lying between 15 and 19 are 16, 17, 18. Out of these, only 17 is a prime number.
           ∴ Given set = {17}, which is not a null set.
       ii) There is no natural number between 1 and 2.
           ∴ Given set is a null set.

11) Which of the following sets are null sets?
       i) A = {x : x² = 16, x  N}
      ii) A = {x : |x| < - 4, x  N}
           i) x² = 16 implies x = ± 4. Out of these, 4  N
              ∴ Given set {4}, which is not a null set.
          ii) The numerical value |x| of any number x cannot be negative.
              ∴ |x| < - 4 is not true for any number
              ∴ Given set is a null set.

12) Which of the following sets are singleton sets?
       i) The set of all prime numbers less than 3.
      ii) {x : |x| = 7, x  N}
          
          i) Natural numbers less than 3 are 1 and 2.
             By definition, 2 is the first prime number.
             ∴ Given set = {2}, which is a singleton
         ii) |x| = 7 implies x = ± 7
             x = -7 does not belong to N, where as x = 7 belong to the set.
             ∴ Given set = {7}, which is a singleton.

13) Which of the following sets are singleton sets?
       i) {x : x² + 2x + 1 = 0, x  N}
      ii) {x : x² = 9, |x| ≤ 3, x  N}
         
         i) x² + 2x + 1 = 0 ⇒ (x + 1)² = 0
            x + 1 = 0 ⇒ x = -1
            x = -1 is not a natural number
            ∴ Given set = {}, which is not a singleton set.
        ii) x² = 9 implies x = ±3
            |x| ≤ 3 implies x = -3, -2, -1, 0, 2, 3.
            ∴ x = -3, 3 satisfies x² = 9 and |x| ≤ 3
            ∴ Given set = {3}, which is a singleton set.