Two sets A and B contains a and b elements respectively. If power set of A contains 16 more elements than that of B, value of ‘b’ and ‘a’ are _______
Answers
Given : Two sets A and B contains a and b elements respectively
power set of A contains 16 more elements than that of B,
To Find : value of ‘b’ and ‘a’
Solution:
Power set is set of all subsets of the given Set including the Set itself and the null set.
Power Set = 2ⁿ
where n = number of elements in given Set
Two sets A and B contains a and b elements respectively
=> Power set of A = 2ᵃ
=> Power set of B = 2ᵇ
power set of A contains 16 more elements than that of B
=> 2ᵃ - 2ᵇ = 16
Let say a = b + x
=> 2ᵇ(2ˣ - 1) = 16
=> 2ᵇ(2ˣ - 1) = 2⁴
RHS has only 2 as factors
LHS has 2ᵇ(2ˣ - 1) where 2ᵇ has factors of 2 but 2ˣ - 1 is odd hence does not have factor so only possibility is :
=> 2ᵇ = 2⁴ and 2ˣ - 1 = 1
=> b = 4 and 2ˣ = 2 => x = 1
a = b + x = 4 + 1 = 5 hence a = 5
a = 5
b = 4
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Answer:
Explanation: 32-16=16, hence a=5, b=4.
Step-by-step explanation:
Explanation: 32-16=16, hence a=5, b=4.