CLASS-8PROBLEM AND SOLUTION OF SUBSET

PROBLEM & SOLUTION OF SUBSETS

1) Find all the Subsets of the set  B = { 0, 1 }

Ans.) The subsets of A containing one element are  { 0 }, { 1 }, also the null set φ and the set B  itself are subsets of B.

So, the subsets of the sets B are φ, { 0 }, { 1 }, { 0, 1 }       (Ans.)

2) Write all the subsets of the set  X = { 1, 2, 3 }

Ans.) The subsets of X containing one element are { 1 }, { 2 }, { 3 }

The subsets of X containing two elements are { 1, 2 }, { 2, 3 }, { 1, 3 }

The null set φ and the set X itself are also subsets of the set A.

So, the subsets of the set X are { φ }, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 2, 3 }, { 1, 3 }, { 1, 2, 3 }               (Ans.)

3) Write all the subsets of the set A = { 1, 2, 3, 4 }

Ans.) The subsets of A containing one element are { 1 }, { 2 }, { 3 }, { 4 }

The subsets of A containing two elements are { 1, 2 }, { 2, 3 }, { 1, 3 }, { 1, 4 }, { 4, 2 }, { 3, 4 }.

The subsets of A containing three elements are { 1, 2, 3 }, { 2, 3, 4 }, { 1, 2, 4 }, { 1, 3, 4 }

The null set φ and the set X itself are also subsets of the set X .

so, the subsets of the set A are { φ }, { 1 }, { 2 }, { 3 }, { 4 }, { 1, 2 }, { 2, 3 }, { 1, 3 }, { 1, 4 }, { 4, 2 }, { 3, 4 }, { 1, 2, 3 }, { 2, 3, 4 }, { 1, 2, 4 }, { 1, 3, 4 }, { 1, 2, 3, 4 }.          (Ans.)

If, X = { 1, 2, 3, 4 } and Y = { 1, 2, 3, 4, 5, 6 } then every element of X is an element of Y but the elements 5 & 6 of Y are not an elements of X.  So,  X ⊂ Y

4) If A = { all the students of Oxford University }

And  X = { all the student of UK } then find the relation

Then we can describe that as, A X .

If X is a finite set containing n elements then the number of proper subset of  X = 2ⁿ - 1.           (Ans.)

5) The proper subsets of the set A = { 1, 2 } are  φ , { 1 } , { 2 }

{ 1, 2 } is a subset of A but not a proper subset,

The number of proper subsets of A = 2² - 1 = 4 – 1 = 3     (Ans.)

6)  If  A = { 1, 2, 3, 4 } and  Q = { x : x – 1,  1< x < 6 , x  N } , then prove  P = Q

Ans.)    A = { 1, 2, 3, 4 }

Q = { x : x – 1,  1< x < 6 , x  N }

As per given the condition x = 2, 3, 4, 5

So substituting the value of  x = { (2 – 1), ( 3 – 1 ), ( 4 – 1 ), ( 5 – 1 ) }

So, the set Q = { 1, 2, 3, 4 }

So, we can say that P = Q             (Ans.)

7) If  A = { 5, 7, 11, 13, 17 } and  B = { 17, 13, 11, 7, 5 }, then prove  A = B

Ans.)  Two sets A & B are equal if each element of A is an element of B and each element of B is an element of A, in other words, The Set A & B are equal if A is a subset of B and B is a subset of A, that is  A    B  and   A .

So, both the sets A & B are equal and denoted by  A = B     (Ans.)

8) Proof P = Q, where P = { letter of the word ROOF } and Q = { letter of the word FOR }

Ans.)  As per the given condition, we can find  P = { letter of the word ROOF } and Q = { letter of the word FOR }

P  = { R, O, F }

Q = { F, O, R }, thus every element of P is a member of Q, or P  Q

Also, every element of Q is a member of P, or Q P,  P = Q    (Ans.)

9) If P = { x / x = 2n + 1, n ∈ N and n < 8 } and Q = { days in a week } then  prove  P  Q

Ans.) As per the given condition P = { x/x = 2n + 1, n ∈ N and n < 8 }   and  Q = { days in a week  }

P = { x / x = 2n + 1, n ∈ N and n < 8 } , here n = 1, 2, 3, 4, 5, 6, 7

So, substituting the value of n in the equation x = 2n + 1

= { ( 2.1 + 1), ( 2.2 + 1 ), ( 2.3 + 1), ( 2.4 + 1 ), ( 2.5 + 1 ), (2.6 + 1 ), ( 2.7 + 1 ) }

=  { 3, 5, 7, 9, 11, 13, 15 }

Here n(p) = 7

and in Q = { days in a week  }

= { Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday }

So,  n(Q) = 7

So, we can observe  n(P) = n(Q)

So,  P ↔  Q              (Ans.)

10)  If P = { x / x = 2n + 1, n ∈ N and n < 8 } and Q = { days in a week } then prove  P = Q

Ans.) As per the given condition P = { x / x = 2n + 1, n ∈ N and n < 8 }  and  Q = { days in a week  }

P = { x / x = 2n + 1, n ∈ N and n < 8 }, here n = 1, 2, 3, 4, 5, 6, 7

So, substituting the value of n in the equation x = 2n + 1

= { ( 2.1 + 1 ), ( 2.2 + 1 ), ( 2.3 + 1 ), ( 2.4 + 1 ), ( 2.5 + 1 ), ( 2.6 + 1 ), ( 2.7 + 1 ) }

=  { 3, 5, 7, 9, 11, 13, 15 }

and in Q = { days in a week }

= { Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday }

Monday to Sunday ∈ Q  but Monday to Sunday ∉ P,

Therefore P ≠ Q            (Ans.)