Questions: How many proper subsets does the set R, colors of the rainbow, have? R= red, orange, yellow, green, blue, indigo, violet Select the correct answer below: 128 64 127 49

How many proper subsets does the set R, colors of the rainbow, have? R= red, orange, yellow, green, blue, indigo, violet Select the correct answer below: 128 64 127 49
Transcript text: CURRENT OBJECTIVE Determine the number of subsets and proper subsets in a given set Question How many proper subsets does the set $R$, colors of the rainbow, have? $R=\{$ red, orange, yellow, green, blue, indigo, violet \} Select the correct answer below: 128 64 127 49 FEEDBACK MORE INSTRUCTION SUBMIT
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Solution

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Solution Steps

To determine the number of proper subsets of a given set, we first need to calculate the total number of subsets. For a set with \( n \) elements, the total number of subsets is \( 2^n \). A proper subset is any subset that is not equal to the original set, so we subtract 1 from the total number of subsets.

Given the set \( R \) with 7 elements (the colors of the rainbow), we can use this approach to find the number of proper subsets.

Solution Approach
  1. Calculate the total number of subsets using \( 2^n \), where \( n \) is the number of elements in the set.
  2. Subtract 1 from the total number of subsets to get the number of proper subsets.
Step 1: Determine the Number of Elements in the Set

The set \( R \) contains the colors of the rainbow: red, orange, yellow, green, blue, indigo, and violet. Therefore, the number of elements in the set is: \[ n = 7 \]

Step 2: Calculate the Total Number of Subsets

The total number of subsets of a set with \( n \) elements is given by: \[ 2^n \] Substituting \( n = 7 \): \[ 2^7 = 128 \]

Step 3: Calculate the Number of Proper Subsets

A proper subset is any subset that is not equal to the original set. Therefore, the number of proper subsets is: \[ 2^n - 1 \] Substituting \( n = 7 \): \[ 2^7 - 1 = 128 - 1 = 127 \]

Final Answer

The number of proper subsets of the set \( R \) is: \[ \boxed{127} \]

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