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

To solve the problem of finding the number of non-empty subsets of set \( A = \{1, 2, 3, 4, \ldots, 10\} \) that contain only even numbers, we can follow these steps:

1. Identify the Even Numbers in the Set:
   The set \( A \) contains the elements \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).
   The even numbers in this set are \( \{2, 4, 6, 8, 10\} \).
2. Determine the Subsets of the Even Numbers:
   The number of subsets of a set with \( n \) elements is \( 2^n \).
   Here, the set of even numbers \( \{2, 4, 6, 8, 10\} \) has 5 elements.
   Thus, the total number of subsets is \( 2^5 = 32 \).
3. Exclude the Empty Subset:
   The question asks for the number of non-empty subsets.
   We exclude the empty subset from the total subsets, so the number of non-empty subsets is \( 32 - 1 = 31 \).

Thus, the number of non-empty subsets containing only even numbers is 31.