0

I already proved the following using proof by induction: For all $a,b,c\in\mathbb{N}$ we have: \begin{align*} a+c=b+c \Rightarrow a=b \end{align*} and \begin{align*} a\cdot c = b\cdot c \Rightarrow a=b . \end{align*}

Now I want to show that for all $a,b,c\in\mathbb{N}$ have that \begin{align*} a=b\Rightarrow a+c=b+c \end{align*} and \begin{align*} a=b\Rightarrow a\cdot c= b\cdot c, \end{align*} but cannot come up with a proper way to do so. Anyone got an idea?

Also got a question about terminology. I know the statements I've already proved are called cancellation law of addition and cancellation law of multiplication, respectively. Is there also a name for the implications $a=b\Rightarrow a*c=b*c$ and the equivalence $a=b\Leftrightarrow a*c=b*c$?

PAT
  • 317

1 Answers1

4

Let us prove by induction on $c$. The proposition clearly holds for $c = 0$: \begin{align*} a = b \Rightarrow 0 + a = 0 + b \end{align*} and we are done.

Now suppose it holds for $c$ and we want to prove it holds for $s(c)$, where $s$ is the successor function.

Moroever, according to its definition as well as the commutative property, one has that \begin{align*} a + c = b + c & \Rightarrow s(a + c) = s(b + c)\\\\ & \Rightarrow s(c + a) = s(c + b)\\\\ & \Rightarrow s(c) + a = s(c) + b\\\\ & \Rightarrow a + s(c) = b + s(c) \end{align*}

Similarly, we can handle the proof for the multiplication.

Let $c = 0$. Then we have that \begin{align*} a = b \Rightarrow 0\cdot a = 0 = 0\cdot b \end{align*} and we are done.

Now suppose that the proposed relation holds for $c$ and we shall prove it holds for $s(c)$ as well.

Since $a = b$, according to the definition of multiplication and its commutative property, it results that

\begin{align*} a\cdot c = b\cdot c & \Rightarrow a\cdot c + a = b\cdot c + b\\\\ & \Rightarrow c\cdot a + a = c\cdot b + b\\\\ & \Rightarrow s(c)\cdot a = s(c)\cdot b\\\\ & \Rightarrow a\cdot s(c) = b\cdot s(c) \end{align*} and we are done.

In order to answer this question, I assumed the definitions of addition and multiplication as given in the book "Analysis I" from Terence Tao as well as the corresponding commutative properties.

Hopefully this helps!

user0102
  • 21,572