Answer to Question #218882 in Discrete Mathematics for Nicholas

Question #218882

Let g be a function from Z+ (the set of positive integers) to Q (the set of rational numbers) defined by (x, y) element of g iff y = (g is a subset of Z+ mapped with Q) and let f be a function on Z + defined by (x, y) element of f iff y = 5x2 + 2x – 3 (f subset of Z+ mapped with Z+)

Which one of the following statements regarding the function g is TRUE? 

(Remember, g is a subset of Z+ mapped with Q.)

1. g can be presented as a straight line graph.

2. g is injective.

3. g is surjective.

4. g is bijective.



1
Expert's answer
2021-07-20T12:36:20-0400

Let gg be a function from Z+Z^+ (the set of positive integers) to QQ (the set of rational numbers) defined by


(x,y)g iff y=4x37 (gZ+×Q)(x, y)\in g\ \text{iff } y=4x-\dfrac{3}{7}\ (g\sube Z^+\times Q)

and let ff be a function on Z+Z^+ defined by


(x,y)g iff y=5x2+2x3 (fZ+×Z+)(x, y)\in g\ \text{iff } y=5x^2+2x-3\ (f\sube Z^+\times Z^+)



We consider the statements provided in the different alternatives:

1. gg is not defined on the set of real numbers thus gg cannot be depicted as a straight line graph.

Only positive integers can be present in the domain of g.g.

It is the case that ordered pairs such as (1,347),(2,747),(3,1147),...(1, 3\dfrac{4}{7}), (2, 7\dfrac{4}{7}), (3,11\dfrac{4}{7}),... belong to gg and these pairs can be presented as dots in a graph. 


2. We prove that gg is indeed injective:


g(u)=g(v)=>4u37=4v37g(u)=g(v)=>4u-\dfrac{3}{7}=4v-\dfrac{3}{7}

=>4(uv)=0=>u=v=>4(u-v)=0=>u=v

gg is injective.


3. The function gg is NOT surjective. 

Counterexample


y=12,12Qy=\dfrac{1}{2}, \dfrac{1}{2}\in Q

Then


y=4x37=12=>4x=1314=>x=1356y=4x-\dfrac{3}{7}=\dfrac{1}{2}=>4x=\dfrac{13}{14}=>x=\dfrac{13}{56}

1356∉Z+\dfrac{13}{56}\not\in Z^+

We conclude that gg is not surjective. 


4. Since gg is not surjective, then gg is not bijective.  



Answer

2. gg is injective.



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment