Math / Discrete

QuestionGiven a set $\mathrm{A}=\{m, p, I, 6, H, G, 5, n\}$ , answer the follc a. Find the number of subsets of $A$ $\square$ b. Find the number of proper subsets of $A$ $\square$

Studdy Solution

STEP 1

What is this asking? How many groups can we make from the items in set A, including the group with nothing in it, and how many if we don't count the group with everything in it? Watch out! Don't mix up subsets and proper subsets!
A proper subset can't have all the elements of the original set.

STEP 2

1. Find the total number of subsets.
2. Find the number of proper subsets.

STEP 3

Let's count the number of elements in set $A$ .
We have $m, p, I, 6, H, G, 5, n$ .
That's $\bf{8}$ elements!

STEP 4

The number of subsets of a set with $n$ elements is given by $2^n$ .
Here, $n$ is $\bf{8}$ , so the number of subsets is $2^{\bf{8}}$ .

STEP 5

$2^8 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = \bf{256}$ .
So, there are $\bf{256}$ subsets of $A$ !

STEP 6

Remember, a proper subset is any subset except the set itself.

STEP 7

Since we already found all the subsets ( $256$ ), we just need to subtract $1$ to remove the set $A$ itself from the count.

STEP 8

$256 - 1 = \bf{255}$ .
So, there are $\bf{255}$ proper subsets of $A$ !

STEP 9

a. There are $\bf{256}$ subsets of $A$ . b. There are $\bf{255}$ proper subsets of $A$ .

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QuestionGiven a set $\mathrm{A}=\{m, p, I, 6, H, G, 5, n\}$ , answer the follc a. Find the number of subsets of $A$ $\square$ b. Find the number of proper subsets of $A$ $\square$

Studdy Solution

STEP 1

What is this asking? How many groups can we make from the items in set A, including the group with nothing in it, and how many if we don't count the group with everything in it? Watch out! Don't mix up subsets and proper subsets!
A proper subset can't have all the elements of the original set.

STEP 2

1. Find the total number of subsets.
2. Find the number of proper subsets.

STEP 3

Let's count the number of elements in set $A$ .
We have $m, p, I, 6, H, G, 5, n$ .
That's $\bf{8}$ elements!

STEP 4

The number of subsets of a set with $n$ elements is given by $2^n$ .
Here, $n$ is $\bf{8}$ , so the number of subsets is $2^{\bf{8}}$ .

STEP 5

$2^8 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = \bf{256}$ .
So, there are $\bf{256}$ subsets of $A$ !

STEP 6

Remember, a proper subset is any subset except the set itself.

STEP 7

Since we already found all the subsets ( $256$ ), we just need to subtract $1$ to remove the set $A$ itself from the count.

STEP 8

$256 - 1 = \bf{255}$ .
So, there are $\bf{255}$ proper subsets of $A$ !

STEP 9

a. There are $\bf{256}$ subsets of $A$ . b. There are $\bf{255}$ proper subsets of $A$ .

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QuestionGiven a set A={m,p,I,6,H,G,5,n}\mathrm{A}=\{m, p, I, 6, H, G, 5, n\}A={m,p,I,6,H,G,5,n}, answer the follc a. Find the number of subsets of AAA □\square□ b. Find the number of proper subsets of AAA □\square□

Studdy Solution

STEP 1

What is this asking? How many groups can we make from the items in set A, including the group with nothing in it, and how many if we don't count the group with everything in it? Watch out! Don't mix up subsets and proper subsets!A *proper* subset can't have *all* the elements of the original set.

STEP 2

1. Find the total number of subsets.2. Find the number of proper subsets.

STEP 3

Let's **count** the number of elements in set AAA.We have m,p,I,6,H,G,5,nm, p, I, 6, H, G, 5, nm,p,I,6,H,G,5,n.That's 8\bf{8}8 elements!

STEP 4

The number of subsets of a set with nnn elements is given by 2n2^n2n.Here, nnn is 8\bf{8}8, so the number of subsets is 282^{\bf{8}}28.

STEP 5

28=2⋅2⋅2⋅2⋅2⋅2⋅2⋅2=2562^8 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = \bf{256}28=2⋅2⋅2⋅2⋅2⋅2⋅2⋅2=256.So, there are 256\bf{256}256 subsets of AAA!

STEP 6

Remember, a *proper* subset is any subset *except* the set itself.

STEP 7

Since we already found *all* the subsets (256256256), we just need to subtract 111 to remove the set AAA itself from the count.

STEP 8

256−1=255256 - 1 = \bf{255}256−1=255.So, there are 255\bf{255}255 *proper* subsets of AAA!

STEP 9

a. There are 256\bf{256}256 subsets of AAA. b. There are 255\bf{255}255 proper subsets of AAA.

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QuestionGiven a set $\mathrm{A}=\{m, p, I, 6, H, G, 5, n\}$ , answer the follc a. Find the number of subsets of $A$ $\square$ b. Find the number of proper subsets of $A$ $\square$

What is this asking? How many groups can we make from the items in set A, including the group with nothing in it, and how many if we don't count the group with everything in it? Watch out! Don't mix up subsets and proper subsets!
A proper subset can't have all the elements of the original set.

1. Find the total number of subsets.
2. Find the number of proper subsets.

Let's count the number of elements in set $A$ .
We have $m, p, I, 6, H, G, 5, n$ .
That's $\bf{8}$ elements!

The number of subsets of a set with $n$ elements is given by $2^n$ .
Here, $n$ is $\bf{8}$ , so the number of subsets is $2^{\bf{8}}$ .

$2^8 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = \bf{256}$ .
So, there are $\bf{256}$ subsets of $A$ !

Remember, a proper subset is any subset except the set itself.

Since we already found all the subsets ( $256$ ), we just need to subtract $1$ to remove the set $A$ itself from the count.

$256 - 1 = \bf{255}$ .
So, there are $\bf{255}$ proper subsets of $A$ !

a. There are $\bf{256}$ subsets of $A$ . b. There are $\bf{255}$ proper subsets of $A$ .