Consider the logic circuit shown in Figure below. The functions $${f_1},$$ $${f_2}$$ and $$f$$ (in canonical sum of products form in decimal notation) are:

$${f_1}\left( {w,\,x,\,y,\,z} \right) = \sum {8,9,10} $$
$${f_2}\left( {w,\,x,\,y,\,z} \right) = \sum {7,8,12,13,18,15} $$
$$f\left( {w,\,x,\,y,\,z} \right) = \sum {\left( {8,9} \right)} $$

The function $${f_3}$$ is

The number of equivalence relations on the set $$\left\{ {1,2,3,4} \right\}$$ is

The determination of the matrix $$$\left[ {\matrix{ 6 & { - 8} & 1 & 1 \cr 0 & 2 & 4 & 6 \cr 0 & 0 & 4 & 8 \cr 0 & 0 & 0 & { - 1} \cr } } \right]\,\,is$$$

Let $$A = ({a_{ij}})$$ be and n-rowed square matrix and $${I_{12}}$$ be the matrix obtained by interchanging the first and second rows of the n-rowed Identity matrix. Then$${AI_{12}}$$ is such that its first