Sets Questions

Prove that the total number of subsets of a finite set containing n elements is \(2^n\).

Solution : Let A be a finite set containing n elements. Let 0 \(\le\) r \(\le\) n. Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is \(^nC_r\). Therefore, the number of subsets of A having …

Prove that the total number of subsets of a finite set containing n elements is \(2^n\). Read More »

Let A and B be two sets containing 2 elements and 4 elements, respectively. The number of subsets A\(\times\)B having 3 or more elements is

Solution : Given, n(A) = 2 and n(B) = 4 \(\therefore\) n(A\(\times\)B) = 8 The number of subsets of (A\(times\)B) having 3 or more elements = \(^8C_3 + {^8C_4} + ….. + {^8C_8}\) = \(2^8 – {^8C_0} – {^8C_1} – {^8C_2}\) = 256 – 1 – 8 – 28 = 219     [\(\because\) \(2^n\) = …

Let A and B be two sets containing 2 elements and 4 elements, respectively. The number of subsets A\(\times\)B having 3 or more elements is Read More »