Power Sets

Question

Consider a set S having exactly n elements; the number of subsets of S that contain at least two elements is given by which expression in terms of n?

RankGuru · Accepted Answer

The total number of subsets of an n-element set is 2^n, since each element is independently included or excluded, a core counting theorem for power sets. Subsets with fewer than two elements are the empty set (1 subset) and the singletons (n subsets). Subtracting these unwanted cases gives 2^n - n - 1 subsets containing at least two elements. This complementary counting approach is a recurring JEE Advanced strategy: count the total and remove forbidden configurations. Option 2^n - n forgets to remove the empty set. Option 2^n - 1 removes only the empty set, leaving all singletons incorrectly included. Option 2^(n-1) counts subsets of even size or half the power set, unrelated to this constraint. Hence the required count is 2^n - n - 1. Plausibility check: for n = 3 the formula gives 8 - 3 - 1 = 4, matching the explicit subsets {a,b},{b,c},{a,c},{a,b,c}, confirming correctness for a small concrete case.

Power Sets

Select the correct option:

🔒 Solution Hidden from View

More sets, relations and functions Practice Questions

A function f from positive reals to reals satisfies f(xy) = f(x) + f(y) for all positive x, y, and f...

If set A has 3 elements and set B has 4 elements, then the number of subsets of the Cartesian produc...

The number of real solutions of the equation given by the absolute value |x - 1| + |x - 3| = 4 over ...

The total number of functions that can be defined from a set with 5 elements to a set with 3 element...

Let f from real numbers to real numbers be defined by f(x) = x^3 + x + 1; which statement about the ...

For the signum function sgn(x), which equals -1, 0, or 1 according to the sign of x, the value of th...

For finite sets the symmetric difference A Δ B is defined as the set of elements in exactly one of A...

If the function f from real numbers to real numbers is defined by f(x) = (3x - 2)/(x + 4) for x not ...

The real function defined by the rule f(x) = the square root of (x - 3) added to the square root of ...

For the real function g(x) = (x^2 + x + 1)/(x^2 + x + 2) defined for every real x, the complete rang...

Which of the following real functions defined for all real x is an odd function according to the sym...

Let [x] denote the greatest integer not exceeding x; the value of the sum [2.7] + [-2.7] + [0.4] + [...

About This Question

Solution