To determine the number of subsets in a set, we use the formula:
\[ \text{Number of subsets} = 2^n \]
where \( n \) is the number of elements in the set.
Here are some questions that are a bit more challenging:
Questions:
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A set \( A \) contains 8 elements. How many subsets does set \( A \) have?
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If set \( B \) consists of the letters in the word "MATH", how many subsets can be formed from set \( B \)?
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Consider a set \( C \) where it has 6 elements and 2 of the elements are repeated. How many unique subsets can be created from set \( C \)?
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A set \( D \) has 10 distinct integers. If we were to remove one integer from set \( D \), how many subsets would the new set have?
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If a set \( E \) contains 12 elements, what would be the number of proper subsets of set \( E \)?
Answers:
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For set \( A \): \[ \text{Number of subsets} = 2^8 = 256 \]
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For set \( B \) (letters in "MATH" are {M, A, T, H}, which has 4 distinct letters): \[ \text{Number of subsets} = 2^4 = 16 \]
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For set \( C \) (assuming it's {a, a, b, c, d, e} which has 5 unique elements {a, b, c, d, e}): To calculate the number of unique subsets: \[ \text{Number of unique subsets} = \frac{2^n}{k!} = \frac{2^5}{2!} = \frac{32}{2} = 16 \] (where \( n \) is the total count of items including repetitions, and \( k \) is the count of the repeated item(s)).
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For set \( D \) (initially has 10 distinct integers):
- Original number of subsets for 10 elements: \[ 2^{10} = 1024 \]
- After removing one integer, set \( D \) would have 9 elements: \[ 2^9 = 512 \]
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For set \( E \) (containing 12 elements): \[ \text{Total number of subsets} = 2^{12} = 4096 \] To find proper subsets, we subtract the set itself: \[ \text{Number of proper subsets} = 4096 - 1 = 4095 \]
Feel free to ask for more problems or any specific area you want to focus on!