A and B are two finite sets. The difference of the number of elements of the power sets is 96 .

step 1 Hello students, welcome back. In this question, we are given the number of elements in A to be e

A and B are two finite sets. The difference of the number of...

Key Concepts

Power Set

A power set is the collection of all subsets of a given set, including the empty set and the set itself. For any finite set, the number of elements in its power set is given by 2 raised to the power of the number of elements in that set, which is a fundamental concept in combinatorics.

Set Cardinality

Set cardinality refers to the number of elements in a set. When dealing with finite sets, cardinality is simply a count of the individual members, and it plays a key role in determining properties such as the size of the power set.

Exponential Functions in Combinatorics

Exponential functions are crucial in counting problems in combinatorics because they describe how quantities such as the number of subsets of a set increase as the set size increases. The formula 2^n demonstrates exponential growth as n increases.

Solving Exponential Equations

Problems that involve differences between values like 2^n often lead to exponential equations. Solving these requires an understanding of the properties of exponents and techniques like factoring and comparing coefficients, which are essential tools in handling combinatorial problems.

Transcript

00:01 Hello students, welcome back.

00:01 In this question, we are given the number of elements in a to be equal to m, okay, and we are given the number of elements in b equal to n.

00:13 Okay, so i have assumed it that number of elements in a is greater than a number of elements in b.

00:19 Okay, i have assumed that to be m and n.

00:22 So, m is greater than n.

00:24 Okay.

00:25 Now we are given the difference between the number of elements in the power set, okay? number of elements number of elements in power set of a will be equal to 2 to the power and why because power set is the set of all the subsets of function of set a right so a total number of subsets of a is 2 to the bar m okay so the number of elements in the power set will be equal to 2 to the bar m similarly the number of elements in the power set of b will be equal to 2 to the part n okay and in this question we are given this value okay that is the difference between they're given the difference between the number of elements in the power set of both these sets to be equal to 96 okay so uh i have just i just have to just have to simplify it if i take two to the par n common out then it becomes two to the par m minus n minus 1 is equal to 96 right so if you see this is an odd part okay this is always old and this part is always even so in the left hand side i have a function which can be split into even and old part so on the right hand side i can also split it up in the odd and even part okay so uh i can write 96 as 32 into 3 okay now this is the old part and this is the even part okay now if you see there is a 2 to the power n form in the even part.

02:04 So i can write 32 in the form of 2 to the power n.

02:07 Right.

02:08 So what i'll write is 2 to the par n times of 2 to the part m minus n minus 1 is equals to 2 to the par 5 into 3...

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