Question

If A =\( \{1, 2, 3, 4, \dots, 10\},\) then the number of non empty subsets of A containing only even number is 
 

• A. 31
• B. 82
• C. 30
• D. 29

Hint:

Always read set theory questions carefully for the word "non-empty" (or "proper subset"). This single word changes the answer by exactly $-1$ and is a very common, easy trap to fall into.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Given $A = \{1,2,3,4,\dots,10\}$.
We need to find the number of non-empty subsets containing only even numbers.
Step 2: Identify Even Elements:
Even elements in $A$ are: \[ E = \{2,4,6,8,10\} \] Number of elements in $E$: \[ n = 5 \]
Step 3: Total Subsets from Even Elements:
Number of subsets of a set with $n$ elements: \[ 2^n = 2^5 = 32 \]
Step 4: Exclude Empty Set:
Since only non-empty subsets are required: \[ \text{Required subsets} = 32 - 1 = 31 \]
Step 5: Final Answer:
\[ \boxed{31} \]