How many proper subsets in all are there of a set with cardrinal number 5?

Answer

To find the number of proper subsets of a set with a cardinal number of 5, we can follow these steps:

### Step 1: Understand the concept of subsets
A subset is a set that contains some or all elements of another set. A proper subset is a subset that contains at least one element but is not identical to the original set.

### Step 2: Calculate the total number of subsets
For any set with cardinal number \( n \), the total number of subsets is given by the formula:
\[ 2^n \]
In this case, since the cardinal number \( n = 5 \):
\[ 2^5 \]

### Step 3: Calculate \( 2^5 \)
Now we calculate \( 2^5 \):
\[ 2^5 = 32 \]
This means there are 32 total subsets of a set with 5 elements.

### Step 4: Calculate the number of proper subsets
The number of proper subsets is equal to the total number of subsets minus the original set itself. Therefore:
\[ \text{Number of proper subsets} = 2^n - 1 \]
Substituting \( n = 5 \):
\[ \text{Number of proper subsets} = 32 - 1 = 31 \]

### Final Answer
Thus, the number of proper subsets of a set with a cardinal number of 5 is:
\[ \boxed{31} \]

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