We can simplify statements in predicate logic using our rules for passing negations over quantifiers, and then applying propositional logical equivalence to the "inside" propositional part. Simplify the statements below $($so negation appears only directly next to predicates$).$ Choose an answer from the drop down menu.

a$.$

$$

$$

x

$$

y

$($

O

$($

x

$)$

$$

$$

$$

$($

y

$)$

$)$

$$ x $$ y $($O$($x$)$E$($y$))$

b$.$

$$

$$

x

$$

$$

y

$$

$($

x

$<$

y

$$

$$

z

$($

x

$<$

z

$$

y

$<$

z

$)$

$)$

$$ x $$ y $($x $<$ y $$ z$($x $<$ z $$ y $<$ z$))$

c$.$ There is a number n for which no other number is either less than n or equal to n$.$

Select an answer

d$.$ It is false that for every number n there are two other numbers which n is between.

Select an answer