**Middle School Number Sense Lesson 108: Sets—Proper & Improper Subsets**

This concept is an extension of the

**SetsSubsets**concept. It appeared

**3 times**this year, with a median placement at

**question # 42**.

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We learned before that a set with

**n**(number of) elements has

**2^n**(number of) subsets. One of these subsets is technically not a

*sub*set at all: the set itself. It is known as the set’s

**improper subset**. There is always

*exactly one*improper subset for every set. Every subset

*other than*the improper subset is considered a

**proper subset**.

So, to figure out how many proper subsets a given set has,

- Count (or determine) the # of elements (
**n**) in the set, and - Use the formula
**2^n – 1**.

**Example 1: {T, E, X, A, S} has ___ improper subset(s)**

- Careful! The question asked for the # of
*improper*subsets. - The answer is always
**1**.

**Example 2: {f, a, c, t, o, r} has ___ proper subsets**

- There are
**6**elements in the set. - 2^6 – 1 = 64 – 1 =
**63**.

**Example 3: The number of elements in a set that has 7 proper subsets is ___**

- Do this calculation backwards. Add 1 (the improper subset) to get
**8**total subsets. - 2^n = 8; n has to be
**3**.

**Example 4: A set with 5 elements has ___ proper subsets**

- The number of elements is given:
**5** - 2^5 – 1 = 32 – 1 =
**31**.

**Example 5: The set {i, n, t, e, g, r, a, l} has ___ proper subsets**

- There are
**8**elements in the set. - 2^8 – 1 = 256 – 1 =
**255**.

**Example 6: The number of elements in a set that has 15 proper subsets is ___**

- Do this calculation backwards. Add 1 (the improper subset) to get
**16**total subsets. - 2^n = 16; n has to be
**4**.

**Here's a free worksheet to help you practice SetsProperImp:**

setsproperimp.pdf |

**Up Next for Middle School: Div25**