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1. Restore the parentheses to these abbreviated propositional forms?

[tex]Q \wedge \backsim S \vee \backsim ( \backsim P \wedge Q )[/tex]

I got this, but am not sure if it is correct.

[tex] [Q \wedge (\backsim S)] \vee ( P \wedge \backsim Q )[/tex]

2. Translate the following with quantifiers.

a. Some isosceles triangle is a right triangle. (all triangles)

This is what I believe it is. [tex](\exists x)(x \ is \ isosceles \ triangle \ \wedge \ x \ is \ a \ right \ triangle)[/tex] Is there anyway to reduce this translation?

b. Between any reall number and any larger real number, there is a rational number (Real numbers)

I have no idea how to write this. Any ideas?

3. Which of the following are true for the universe of all real numbers

(a) [tex] (\forall x)(\exists y)(x \leq y) [/tex]

I said true, because for all x's, there exists a y to fit this equation

(b) [tex](\forall x)(\exists!y)(x=y^2)[/tex]

what does [tex](\exists!y) [/tex] even mean?

Lastly,

4. Let the function a(x) be an open sentence with variable x.

Prove that [tex](\exists!x)A(x) \Rightarrow (\exists x)A(x)[/tex]

How to do this?

Any help would be greatly appreciated. Thanks!