Set Theory

If and the total number of possible relations from set to set is 1024, then is:

The number of relations from set A to set B is given by . We are given that and the number of relations is 1024. So, we have:

Since , we have:

Therefore, , which means .

Answer: (D) 5

If , then the number of subsets of which contain at least two elements is:

The total number of subsets of A is . The number of subsets with at least two elements is equal to the total number of subsets minus the number of subsets with zero elements (the empty set) and the number of subsets with one element.

Number of subsets with zero elements = 1 (the empty set)

Number of subsets with one element = 6 (one for each element)

Number of subsets with at least two elements =

Let be the total number of families in the town.

Let be the set of families who own cellphones.

Let be the set of families who own scooters.

From the problem statement, we have:

• The number of families owning cellphones is of , which is .
• The number of families owning scooters is .
• The number of families owning both cellphones and scooters is of , which is .

We are told that every family owns at least one of the two items. This means the union of the sets and is equal to the total number of families :

We apply the Principle of Inclusion-Exclusion for two sets:

Now, substitute the known values into the formula:

Combine the terms involving on the right side:

Subtract from both sides of the equation:

Solve for by dividing both sides by :

Therefore, the total number of families in the town is 30,000.

The correct option is (B) 30000.

Let the first set have elements, and the second set have elements. The number of subsets of a set with elements is given by .

The number of subsets of the first set is .
The number of subsets of the second set is .

We are given that the total number of subsets of the first set is 56 more than the total number of subsets of the second set. This can be written as:

We can rewrite this equation as:

We can factor out :

Since , we can consider the factors of 56 which are powers of 2. The factors are 1, 2, 4, 8, 14, 28, 56.
We can write 56 as , where is an integer.
Then .

If , then .
In this case, , which gives .
Thus, , so .

If , then . This gives , so , which is not possible since must be a power of 2.
If , then . This gives , so , which is not possible.
If , then . This gives , so , which is not possible.
If , then this is not a power of 2.
If , then this is not a power of 2.

Therefore, the only solution is and .

Final Answer: The final answer is .