a$)$ We can draw Truth Trees for a single sentence as well as for an

argument. Draw a truth tree for the sentence displayed below. $(\text{'}$S $(y,x)\text{'}$ is

short for $\text{'}y$ is the successor of $x^{\text{'}}$ or $\text{'}y$ is one greater than $x^{\text{'}}\text{'}).$

AAxEEyS$(y,x)$

Whenever you get a new name in your proof,

instantiate the Universal Quantifier with that new name, and

continue the tree. Be careful. Make sure that the tree contains an

instance of every existential generalization. The tree should be peculiar. I don't

want the whole tree, $1$ just want enough to see that you understand.

b$)$ suppose you had an argument with a truth tree that looked peculiar in the

same way the tree for this sentence looks peculiar. Is the argument proved

$($and valid$)$ or not proved $($and invalid$)?$

c$)$ Suppose you gave an interpretation that made every sentence of your tree

true. Consider the Universe of discourse, also called the 'Domain' $-$ all the

things the interpretation says exists. What can you say about it$?$