In class we showed that $\{uarr\}$ is functionally complete by starting from the fact that multiplication, complement $\}$ is functionally complete and then expressing both of those operations using uarr alone.

$($a$)(2$ points$)$ Show that uarr $($the NAND operation$)$ is not associative by finding three bits $a,b,c$ such that $(auarrb)uarrc$auarr$(buarrc).$

$($b$)(2$ points$)$ Express the Boolean addition $x+y$ in terms of uarr alone.

$($c$)(3$ points$)$ Write a Boolean function $f$:$\{0,1{\}}^2\{0,1\}$ that uses uarr alone to produce the behavior in this input$/$output table. Note that since uarr is not associative, be very careful about your parentheses! Hint: Start by writing $f$ in sum of minterms form.

$\backslash$table$[[x,y,f(x,y)$