All the 32 subsets for (x,y,z,u,v)
No shortcuts here. Just start listing them, in order of number of elements:
{}
{x},{y},...
{x,y},{x,z},...
{x,y,z},{x,y,u},...
{x,y,z,u},{x,y,z,v},...
{x,y,z,u,v}
To find all the subsets of the set (x, y, z, u, v), we can use the concept of binary representation. Each element in the set can either be included or excluded from a particular subset, and this can be represented as a binary digit (1 for inclusion, 0 for exclusion).
Since the set has 5 elements, we need 5 binary digits to represent all the possible subsets. We can start by listing all the numbers from 0 to 31 (which is 2^5 - 1), and for each number, convert it to its binary representation of length 5.
Here are the 32 subsets for the set (x, y, z, u, v):
Subset 1: (0, 0, 0, 0, 0)
The binary representation is 00000, which means none of the elements are included.
Subset 2: (0, 0, 0, 0, 1)
Binary representation: 00001, which means only the element v is included.
Subset 3: (0, 0, 0, 1, 0)
Binary representation: 00010, which means only the element u is included.
Subset 4: (0, 0, 0, 1, 1)
Binary representation: 00011, which means u and v are included.
Subset 5: (0, 0, 1, 0, 0)
Binary representation: 00100, which means only the element z is included.
Subset 6: (0, 0, 1, 0, 1)
Binary representation: 00101, which means z and v are included.
Subset 7: (0, 0, 1, 1, 0)
Binary representation: 00110, which means z and u are included.
Subset 8: (0, 0, 1, 1, 1)
Binary representation: 00111, which means z, u, and v are included.
Subset 9: (0, 1, 0, 0, 0)
Binary representation: 01000, which means only the element y is included.
Subset 10: (0, 1, 0, 0, 1)
Binary representation: 01001, which means y and v are included.
...and so on...
Subset 32: (1, 1, 1, 1, 1)
Binary representation: 11111, which means all the elements, x, y, z, u, and v, are included.
By generating all the binary numbers from 0 to 31 and converting them to their respective subsets, we have found all 32 subsets for the set (x, y, z, u, v).