$6.$ Use quantified predicates to express the following sentences.

a$.$ The sum of two negative integers will always be negative.

b$.$ For each positive integer, there doesn$$t exist any negative integer that has the same value as it$.$

c$.$ For any two integers, there is always another integer equals to their $($the first two integers$)$ product.

$7.$ Negate each predicate you created in question $6,$ then simplify them to remove the negation operators. While

simplifying, clarify which logic rule$($s$)$ you use in each step.

$8.$ For each sub$-$question, please answer the following questions. Is state $$ proper for proposition $/$ predicate $?$ If

no$,$ explain why; if yes, does $$ satisfy $?$

a$.=\{=\},(->)(->)$

b$.=\{=,=\},=\backslash$cup $\{=\},$

c$.=\{=5,=0,=6\},>[]$

d$.=\{=5,=(5,8)\},+1=[0]$

e$.=\{=,=\},$ where $=(2,0,4),[[1]]=2$

$9.$ Find all states $($containing only bindings for $,$ and $)$ such that $.$ Briefly explain each state

you found.

$10.$ Define predicate functions.

a$.$ Define predicate function $(,,)$ which returns $$ if and only if positive $$ is not larger than

the length of array $,$ and integer $$ is greater than each of the first $$ numbers in $.$ For example, under

state $=\{=(2,4,1,6)\},(,3,5)$ returns $,(,3,4)$ returns $.$ Remind

that, you can use $()$ to find the length of array $.$

b$.$ Define predicate function $(,)$ which returns $$ if and only if every integer in array $$ is

greater than some integer in array $.$ For example, $((4,5,2,3),(8,1,5,8))$ returns $.$