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Let S be a set which has N elements.

Consider in how many ways we can choose a subset.

List the N elements of the set S.

Let the names of the N elements be,

x1, x2, x3, . . . xN

For an arbitrary subset,

we have two choices for x1.

Namely, x1 might or might not be in the subset.

We have two choices for x2.

Namely, x2 might or might not be in the subset.

We have two choices for x3.

Namely, x3 might or might not be in the subset.

. . .

We have two choices for xN.

Namely, xN might or might not be in the subset.

Now we can easily count the total number of ways to choose

a subset.

2 choices for x1 times 2 choices for x2 times . . .

= 2 to the Nth power choices of ways to choose a subset.

This proves that the number of subsets of a set with

N elements is 2 raised to the Nth power.

Kermit Rose

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Q: WHAT IS THE Proof that set N has 2 POWER N subset?
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How many subset can you form in algebra?

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How many subsets does a set have if the set has four elements?

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