MHB Prove that the statement is always true using the rules of boolean algebra

AI Thread Summary
The discussion revolves around proving the statement {b ∧ [¬a ⇒ ¬b]} ⇒ a using boolean algebra, with participants clarifying that the implication symbol (⇒) is not part of boolean algebra's operational symbols. There is confusion regarding the correct interpretation of the statement, with suggestions that it may contain a typo. Participants explore the transformation of the statement into a form that can be analyzed, ultimately concluding that the original statement is not always true. The conversation highlights the distinction between boolean algebra and propositional calculus, emphasizing that the latter can be used to prove the formulas in boolean algebra.
mathmari
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Hey! 😊

I want to prove by using the rules of boolean algebra that the following statement is always true $$\{b\land [\neg a\Rightarrow \neg b]\} \Rightarrow a$$

Since we have to use the rules of boolean algebra, we cannot use the truth table, right?

Could you give me a hint how we could show that?

:unsure:
 
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Note that $p\to q$ is the same a $\lnot p\lor q$.

Btw, can it be that there is a typo? Can it be that it should be $\{b, [\neg a\Rightarrow \neg b]\} \Rightarrow a$? 🤔
 
Klaas van Aarsen said:
Note that $p\to q$ is the same a $\lnot p\lor q$.

Btw, can it be that there is a typo? Can it be that it should be $\{b, [\neg a\Rightarrow \neg b]\} \Rightarrow a$? 🤔

Before the statement there is the following text:

Proofs of contradiction often have the following pattern:
You know that a statement $b$ is correct, and would like to show that a statement $a$ is also correct. We consider then that $a$ is not correct, and show that it follows that $b$ is wrong (“contradiction”), and then we conclude that $a$ is correct. So the given statement is correct, isn't it? :unsure:

We have that \begin{align*}\{b\land [\neg a\Rightarrow \neg b]\} \Rightarrow a &\equiv \{b\land [a\lor \neg b]\} \Rightarrow a \\ & \equiv \{b\land a\lor b\land \neg b]\} \Rightarrow a \\ & \equiv \{b\land a \}\Rightarrow a \\ & \equiv \neg \{b\land a \}\lor a \\ & \equiv \neg b\lor \neg a \lor a \\ & \equiv \neg b\end{align*}

Is everything correct? :unsure:
 
Ah, I assumed those curly braces were intended to indicate a set of boolean statements.
I guess it's also possible that they are intended as alternatives to parentheses.
In that case, I believe it is correct.
It also means that the statement is not always true...


EDIT: the last step is not correct as SquareOne pointed out. 🤨
 
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\[ \{ b \land [\neg a \to \neg b] \} \to a \\ \{ b \land [a \lor \neg b] \} \to a \\ \{ [b \land a] \lor [b \land \neg b] \} \to a \\ \{ [b \land a] \lor \text{False} \} \to a \\ [b \land a] \to a \\ \neg [b \land a] \lor a \\ \neg b \lor \neg a \lor a \\ \neg b \lor \text{True} \\ \text{True} \]
 
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Ohh I see! Thanks a lot! (Sun)
 
SquareOne said:
\[ \{ b \land [\neg a \to \neg b] \} \to a \\ \{ b \land [a \lor \neg b] \} \to a \\ \{ [b \land a] \lor [b \land \neg b] \} \to a \\ \{ [b \land a] \lor \text{False} \} \to a \\ [b \land a] \to a \\ \neg [b \land a] \lor a \\ \neg b \lor \neg a \lor a \\ \neg b \lor \text{True} \\ \text{True} \]

This is not a proof by contradiction using boolean algebra
In boolean algebra the symbol $\rightarrow$ does not exist
However it exists in propositional calculus
The operational symbols in Boolean algebra are: $\wedge...\vee$...,(') for complement of a boolean variable x i.e x'
and the constants T or F or better 0 or 1
And of course the pridicate of equality
The axioms you can find in any book or inthe internet
 
mathmari said:
Before the statement there is the following text:

Proofs of contradiction often have the following pattern:
You know that a statement $b$ is correct, and would like to show that a statement $a$ is also correct. We consider then that $a$ is not correct, and show that it follows that $b$ is wrong (“contradiction”), and then we conclude that $a$ is correct. So the given statement is correct, isn't it? :unsure:

We have that \begin{align*}\{b\land [\neg a\Rightarrow \neg b]\} \Rightarrow a &\equiv \{b\land [a\lor \neg b]\} \Rightarrow a \\ & \equiv \{b\land a\lor b\land \neg b]\} \Rightarrow a \\ & \equiv \{b\land a \}\Rightarrow a \\ & \equiv \neg \{b\land a \}\lor a \\ & \equiv \neg b\lor \neg a \lor a \\ & \equiv \neg b\end{align*}

Is everything correct? :unsure:

Proof by contradion for the statement $b\Rightarrow a$ and following the pattern that you mentioned above is the following formula $[b\wedge(\neg a\Rightarrow\neg b)]\Rightarrow (b\Rightarrow a)$ and not the statement
\begin{align*}\{b\land [\neg a\Rightarrow \neg b]\} \Rightarrow a\end{align*}
And that statement (your stetement ) can be proved only by using propositional calculus and not boolean algebra
Boolean algebra does not include $\Rightarrow$ as an operational symbol
 
Klaas van Aarsen said:
Note that $p\to q$ is the same a $\lnot p\lor q$.

Btw, can it be that there is a typo? Can it be that it should be $\{b, [\neg a\Rightarrow \neg b]\} \Rightarrow a$? 🤔
does boolean algebra include $\rightarrow$ as one of its operational symbols
You can use logic (propositinal and predicate calculus) to prove all the formulas in boolean algebra but not the opposite
 
  • #10
mathmari said:
Hey! 😊

I want to prove by using the rules of boolean algebra that the following statement is always true $$\{b\land [\neg a\Rightarrow \neg b]\} \Rightarrow a$$

Since we have to use the rules of boolean algebra, we cannot use the truth table, right?

Could you give me a hint how we could show that?

:unsure:
This is not a formula of contradiction:
All formulas of contradiction are the following:

1)$(\neg b\rightarrow(a\wedge \neg a))\rightarrow b$

2)$(\neg b\rightarrow b)\rightarrow b$

3)$(b\rightarrow\neg b)\rightarrow\neg b$

4)$[(b\wedge\neg a)\rightarrow (r\wedge\neg r)]\rightarrow(b\rightarrow a)$

5) $[(b\wedge\neg a)\rightarrow \neg b)]\rightarrow(b\rightarrow a)$

6) )$[(b\wedge\neg a)\rightarrow a]\rightarrow(b\rightarrow a)$

You can check all the above formulas by using propositional calculus
 

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