# Let S be a set consisting of 10 elements. The number of tuples of the form (A, B) such that A and B are subsets of S, and A ⊆ B is _______

143 views
in Algebra
closed
Let S be a set consisting of 10 elements. The number of tuples of the form (A, B) such that A and B are subsets of S, and A ⊆ B is _______

by (41.6k points)
selected by

Explanation:

Let’s take an example of 4 element set

{ a, b, c, d}

Now we need to find “tuples of the form (A,B) such that A and B are subsets of S”.

• Let’s take A as {a} , now calculate how many B’s are possible such that  AB.
 ({a} , {a}) ({a} ,{a, b}) ({a} , {a, c}) ({a} , {a, d}) ({a} , {a, b, c}) ({a} , {a, c, d}) ({a} , {a, b, d}) ({a} , {a, b, c, d})

8 tuples are possible.

Now if we take Φ as A then total 16 tuples are possible . and

If we take {a, b} as A then total 4 tuples are possible and for A we can chose nC2  ways so total nC2 × 4 tuples are possible for 2 element in A. And so on we can calculate for 2 element subsets and 3 element subsets.

So its general form for no of tuples possible (if n elements are given :

nC0×(2n)  + nC1×(2n-1) + nC2×(2n-2) + ...+ nCn×(20) = (2+1)n

Calculation :

We have given number of elements as 10 so total number of tuples will be (2+1)10 = 310 = 59049.