Question

Determine whether each statement is true or false, and prove or disprove, as appropriate.

(a) (∀x∈R)(∃y∈R)[xy=1].(∀x∈R)(∃y∈R)[xy=1].

(b) (∃x∈R)(∀y∈R)[xy=1].(∃x∈R)(∀y∈R)[xy=1].

(c) (∃x∈R)(∀y∈R)[xy>0].(∃x∈R)(∀y∈R)[xy>0].

(d) (∀x∈R)(∃y∈R)[xy>0].(∀x∈R)(∃y∈R)[xy>0].

(e) (∀x∈R)(∃y∈R)(∀z∈R)[xy=xz].(∀x∈R)(∃y∈R)(∀z∈R)[xy=xz].

(f) (∃y∈R)(∀x∈R)(∃z∈R)[xy=xz].(∃y∈R)(∀x∈R)(∃z∈R)[xy=xz].

(g) (∀x∈Q)(∃y∈Z)[xy∈Z].(∀x∈Q)(∃y∈Z)[xy∈Z].

(h) (∃x∈Z+)(∀y∈Z+)[y≤x].(∃x∈Z+)(∀y∈Z+)[y≤x].

(i) (∀y∈Z+)(∃x∈Z+)[y≤x].(∀y∈Z+)(∃x∈Z+)[y≤x].

(j) (∀x,y∈Z)[x<y⇒(∃z∈Z)[x<z<y]].(∀x,y∈Z)[x<y⇒(∃z∈Z)[x<z<y]].

(k) (∀x,y∈Q)[x<y⇒(∃z∈Q)[x<z<y]].(∀x,y∈Q)[x<y⇒(∃z∈Q)[x<z<y]].

Answer 1

(a).  Given statement ,

If we choose that for any real number , .

So the exist real number x such that for all real number y , .  

Hence the given statement is FALSE .

(b). Given statement

If the statement is true then for that fixed x , if we choose y = 0 then ,

, a contradiction .

Hence the statement is FALSE .

(c). Given statement ,

If the statement is true then for that x if we choose y= 0 as this is true for all choise of y then ,

, a contradiction .

Hence the statement is FALSE .

(d). Given statement ,

If we choose x =0 then the is no real number y such that as.

So there is a real number x such that does not holds .

Hence the statement is FALSE.