Question

a relation defined in a non empty set A having n elements has​

Answer 1

Step-by-step explanation:

How many non empty relation can be defined on a set A having 2 elements? ... Since A x A has 4 elements. So there will be 2^n subsets , where n = the number of elements.

Answer 2

Answer:

Final Answer.

Step-by-step explanation:

Given,

A non empty set A contains n elements.

Using Mathematical Induction,

Suppose,

∣A∣ = 0

Then,

A = ∅

But we know that an empty set is only a subset of itself.

So,

∣P(A)∣ = 1 = $2^{0}$

Now, let's say |A| = n.

So, by induction method we know that ∣P(A)∣ = $2^{n}$ → 1.

Let, B be a set with (n+1) elements,

then, B = A ∪ {a}

Now, there are 2 kinds of subsets of B: those that include 'a' and those that don't. The first ones are exactly the subset of X which do not contain 'a' and there are $2^{n}$ of them.

The second ones are of the form C ∪ {a}, where C ∈ P(A).

Since, there are $2^{n}$ possible choices of C, there must be exactly $2^{n}$ subsets of B of which 'a' is an element.

Therefore,

∣P(B)∣ = $2^{n}$ + $2^{n}$ = $2^{n+1}$.

Hence, if a set has n elements then the power set has $2^{n}$ elements.

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