Number of subsets of a set of $n$ elements $ = 2^n$
$\quad = {}^nC_0 + {}^nC_1+\ldots + {}^nC_n$
Each of these terms ${}^nC_k$ denotes the number of possible subsets of size $k.$
Now given a subset of size $k,$ how many subsets can it have? $\implies 2^k.$
So, in this way number of inclusive relations on subsets of a set with $n$ elements
$\quad = 2^0 \times {}^nC_0 +2^1 \times {}^nC_1+\ldots +2^n \times {}^nC_n = 3^n$
PS: $3^n = (2+1)^n = {}^nC_0 \times2^0\times1^n + {}^nC_1\times 2^1 \times1^{n-1} \ldots {}^nC_n\times 2^n\times1^0$