Answered: Produce by hand a proof of ((A→B)→C) →…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Produce by hand a proof of ((A→B)→C) → (A→(B→C))

How many lines does the most natural box proof contain?

Expert Solution

Step 1

Let us first find the truth table of $(A \to B) \to C$ and $A \to (B \to C)$ .

For $(A \to B) \to C$ :

$A$	$b$	$c$	$a \rightarrow b$	$\left(a \rightarrow b\right) \rightarrow c$
True	True	True	True	True
True	True	False	True	False
True	False	True	False	True
True	False	False	False	True
False	True	True	True	True
False	True	False	True	False
False	False	True	True	True
False	False	False	True	False

And for $A \to (B \to C)$ :

$A$	$b$	$c$	$\to C$	$a \rightarrow \left(b \rightarrow c\right)$
True	True	True	True	True
True	True	False	False	False
True	False	True	True	True
True	False	False	True	True
False	True	True	True	True
False	True	False	False	True
False	False	True	True	True
False	False	False	True	True

It can be observed that $(A \to B) \to C$ and $A \to (B \to C)$ are not equivalent. So, the proof by truth table will be more convenient.

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