Tao Analysis I - A.7.1

Exercise A.7.1

Suppose you have four real numbers $a$, $b$, $c$, $d$ and you know that $a = b$ and $c = d$. Use the above four axioms to deduce that $a + d = b + c$.

Let's remind ourselves of the four equality axioms presented in the book:

• Reflexivity. $x=x$.
• Symmetry. If $x=y$, then $y=x$.
• Transitivity. If $x=y$ and $y=z$, then $x=z$.
• Substitution. If $x=y$, then $f(x)=f(y)$ for all functions or operations $f$. Similarly if $x=y$, then $P(x)=P(y)$ for property $P$.
  

To answer the question, we have to try hard not to fall back to school algebra. The challenge is to go back to fundamentals and only use the axioms given.

Let's start with the given facts.

$$\begin{align} a &= b \\ c &=d \end{align}$$

Now let's use substition on $a=b$ using the function $f(x) = x + d$. This gives us

$$\begin{align} f(a) &= f(b) \\ a + d &=  b + d \end{align}$$

We can use substitution on $c=d$ using the function $g(x) = b + x$. This gives us

$$\begin{align} g(c) &= g(d) \\ b + c &=  b + d \end{align}$$

In preparation for the next step, we use symmetry to say

$$b + c =  b + d \implies b +d = b + c$$

Finally we use transitivity,

$$  (a + d =  b + d) \land (b + d =  b + c)  \implies (a+d = b+c) $$

The proof is complete. $\square$