1. ## Help with implications

I want to prove that

$\displaystyle a < b \Rightarrow a + c = b + c$

Which of course is not true.

I don't know how to do this kind of proofs.
I mean, when trying to prove
$\displaystyle A \Rightarrow B$
should I try to work on the left side of the implication until A is equal to B? Am I allowed to take on what I have on the left side and make a substitution of that on the right side?

i.e., is this allowed?:
$\displaystyle a < b \Leftrightarrow a + x = b, x > 0$

$\displaystyle a < b \Rightarrow a + c = b + c$
$\displaystyle a + x = b \Rightarrow a + c = b + c$
$\displaystyle a + x = b \Rightarrow a + c = (a + x) + c$

2. Originally Posted by devouredelysium
I want to prove that

$\displaystyle a < b \Rightarrow a + c = b + c$

Which of course is not true.
Then what do you mean "I want to prove" it? You can't prove something that isn't true!

I don't know how to do this kind of proofs.
I mean, when trying to prove
$\displaystyle A \Rightarrow B$
should I try to work on the left side of the implication until A is equal to B?
I don't know what you mean by "A is equal to B". In general, the hypothesis and conclusion of a proof (A and B) are statements or combinations of statements and it doesn't make sense to say "equal".

Am I allowed to take on what I have on the left side and make a substitution of that on the right side?

i.e., is this allowed?:
$\displaystyle a < b \Leftrightarrow a + x = b, x > 0$

$\displaystyle a < b \Rightarrow a + c = b + c$
$\displaystyle a + x = b \Rightarrow a + c = b + c$
$\displaystyle a + x = b \Rightarrow a + c = (a + x) + c$
I have no idea what you are trying to do here.

If you want to prove the "a< b" implis "a+ x= b for x> 0" then somewhere you are going to have to use the definition of "a< b". How is that defined in your course?

One thing that it is important to realize- definitions in mathematics are working definitions- you use the precise words of definitions in proofs.

3. Originally Posted by devouredelysium
I want to prove that

$\displaystyle a < b \Rightarrow a + c = b + c$

Which of course is not true.

I don't know how to do this kind of proofs.
I mean, when trying to prove
$\displaystyle A \Rightarrow B$
should I try to work on the left side of the implication until A is equal to B? Am I allowed to take on what I have on the left side and make a substitution of that on the right side?
I don't understand why you want to prove something that you know is false, but I know mathematically to prove $\displaystyle P \Rightarrow Q$ you go from P to Q, and to prove $\displaystyle P \Leftrightarrow Q$, you must prove $\displaystyle P \Rightarrow Q$ and $\displaystyle Q \Rightarrow P$

To prove $\displaystyle a < b \Rightarrow a+c = b+ c$ is false.

From truth table of implication, we know that if $\displaystyle P$ is true and $\displaystyle Q$ is false, the implication $\displaystyle P \Rightarrow Q$ is false. Based on this logical principle, we proceed as follows:

Let $\displaystyle a,b \in R$. By Trichotomy Law, if $\displaystyle a,b \in R$, then there are three possiblilities, namely $\displaystyle a<b, a=b$, or $\displaystyle a>b$.

Suppose that $\displaystyle a<b$. Then $\displaystyle a\not < b$ and a $\displaystyle \not =b$.

Case 1: Say $\displaystyle a\not =b$. Adding $\displaystyle c$ to bothside of equality, by principle of arithmetic operation $\displaystyle a+c \not= b+c$, which in logical term $\displaystyle a<b \Rightarrow a+c\not=b+c$ as desired.

Case 2: $\displaystyle a\not >b$. The process is similar.

It's not an exciting proof.