Question about Mathematical Logic Implication

[ALine]

I am attempting to understanding implication in mathematical logic.

I will start off by giving an example of its use and my interpretation of how to correctly under it. That way my process can be critiqued and corrected.

Example

if "f(a) =f(a')", then " a = a' "

Interpretation

the given logical statement above is like an "instruction manual" in which if I have f(some a) = f(some b) then it will always result in some a equaling some a'.

And if I observe a counter example to this then that would mean that the overall example statement is wrong and therefore it is not defined as being that said thing, in this case

being the definition of a one to one function.

That being said if I have the function

f = {(1, a), (2, a), (3, b)} and I do some searching I find that f(1) = f(2). Because of this I can use the above example logical statement.

if f(1) = f(2), then 1 = 2. However this is not the case and therefore the initial statement is thereby false and thus it is not a one to one function.

 

Is this interpretation correct?

Also is this "instructional following" a good method of interpretation mathematical logic?

Thank you for your future answers.

[joigus]

Down to basics, I think you're sniffing around the idea that,

(A ==> B) <==> (Neg B ==> Neg A)

For the particular statement,

A: f(x)=f(y)

B: x=y

So if you find any occurrence of x=/=y (Neg B) and f(x)=f(y) (A); you've proven f not to be injective or one-to-one.

Is that it?

[ALine]

30 minutes ago, joigus said:

Down to basics

yep! I just changed my major last semester and am taking discrete next. I got the introductions of proofs and a little discrete, however did not go that deep.

Spending the summer studying discrete, linear algebra, and vector analysis so that I can be prepared when classes start back up again.

31 minutes ago, joigus said:

(A ==> B) <==> (Neg B ==> Neg A)

For the particular statement,

A: f(x)=f(y)

B: x=y

So if you find any occurrence of x=/=y (Neg B) and f(x)=f(y) (A); you've proven f not to be injective or one-to-one.

Is that it?

yes and no. yes I believe that this is what I am trying to say and convey, no because I want to understand it intuitively. I am a novice when it comes to this stuff, however I am really good at abstraction so I keep up

[ahmet]

19 hours ago, ALine said:

I am attempting to understanding implication in mathematical logic.

I will start off by giving an example of its use and my interpretation of how to correctly under it. That way my process can be critiqued and corrected.

Example

if "f(a) =f(a')", then " a = a' "

Interpretation

the given logical statement above is like an "instruction manual" in which if I have f(some a) = f(some b) then it will always result in some a equaling some a'.

And if I observe a counter example to this then that would mean that the overall example statement is wrong and therefore it is not defined as being that said thing, in this case

being the definition of a one to one function.

That being said if I have the function

f = {(1, a), (2, a), (3, b)} and I do some searching I find that f(1) = f(2). Because of this I can use the above example logical statement.

if f(1) = f(2), then 1 = 2. However this is not the case and therefore the initial statement is thereby false and thus it is not a one to one function.

 

Is this interpretation correct?

 

if you meant the derivation function,then your implication is incorrect!

because  [math] f(2x)=f((x^{2}+c)')  [/math] but   [math] 2x \ne x^{2}+c [/math]

if you meant whether  1-1 is equivalent of this proposition (given below) ,then yes!

"f(x)=f(y) --->> x=y"

but if you are asking whether ,being well defined is equivalent meaning of 1-1 ,then no.

let see one example given in the figure.

Edited June 8, 2020 by ahmet

[ALine]

I think what I am trying to understand is if my interpretation is correct from a beginners perspective. Like if my step my step process is correct vs. if the overall thing is correct. Analogous to trying to initially understand a physics equation intuitively verses begin able to solve it. Thank you both however for the assistance.

Edited June 8, 2020 by ALine
change "t" to "to"

[ahmet]

3 minutes ago, ALine said:

I think what I am trying to understand is if my interpretation is correct from a beginners perspective. Like if my step my step process is correct vs. if the overall thing is correct. Analogous to trying to initially understand a physics equation intuitively verses begin able to solve it. Thank you both however for the assistance.

I understand. physics and mathematics are different disciplines. 

as a matheatician,to me,if you would like to do mathematics ,then you need to concentrate on definitions and theorems.

[joigus]

1 hour ago, ALine said:

yes and no.

"Yes and no" doesn't bode well with standard logic.

Yes and no is the empty set of propositions. As soon as you get out of the empty set, give me a call. Maybe I can help.

11 minutes ago, ahmet said:

if you meant the derivation function,then your implication is incorrect!

He doesn't mean anything remotely similar to the derivation function.

12 minutes ago, ahmet said:

if you meant whether  1-1 is equivalent of this proposition (given below) ,then yes!

"f(x)=f(y) --->> x=y"

That's more like what they mean.

13 minutes ago, ahmet said:

but if you are asking whether ,being well defined is equivalent meaning of 1-1 ,then no.

At no point does @ALine write the words "well defined." This is your imagination speaking.  

You seem to assume lots of things that neither you nor others have written about.  

1 hour ago, ALine said:

I want to understand it intuitively.

Say you want to see how,

(A ==> B) <==> (Neg B ==> Neg A)

A: I'm Italian

B: I'm European

Obviously: A ==> B (if I'm Italian, them I'm European)

Let's see that this is equivalent to Neg B ==> Neg A

I'm not European ==> I'm not Italian

Is that intuitive enough?

[ALine]

6 minutes ago, joigus said:

Yes and no is the empty set of propositions. As soon as you get out of the empty set, give me a call. Maybe I can help.

ha, I think need to do more work before I can get out of the "mutually exclusive" club.

Let me go back and see if I can better refine my question so that I can better convey it to everyone.