$3.$ Additional Operators $(7$ points$)$

Consider this NAND operator $,$ which is defined as follows p $$ q $($p $$ q$).$

We haven$$t seen this operator in our class lecture but it works like NOT$-$AND.

$(1)(3$ points$)$ Using truth tables show that $$ is associative,

i$.$e$.,$ p $($q $$ r$)($p $$ q$)$ r$.$

p q r $($q $$ r$)($p $($q $$ r$))($p $$ q$)(($p $$ q$)$ r$)($p $($q $$ r$))(($p $$ q$)$ r$)$

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

$(2)(4$ points$)$ Using truth tables show that $$ is not associative,

i$.$e$.,$ p $($q $$ r$)6($p $$ q$)$ r$.$