I need to write the following sentence as a formula in propositional logic: Proposition a is false, if the two propositions b and c have different truth values; otherwise a is true.
This is how I solved it:
$$ a \lor (b \land c)$$
Is it correct? If my solution is wrong, what would the correct option be? I am a beginner and I am trying to learn this on my own so I have no one else to ask, don't be to harsh on me please!
Your attempt is incorrect. You have $$a \vee (b \wedge c)$$ This is true is $a$ is true or if $b$ and $c$ are both true. This is not at all what is asked for. The proposition is that $a$ is false, if and only if $b$ and $c$ have different values. This is the same as saying that $a$ is true if and only if $b$ and $c$ have the same truth value, right?
This $a \leftrightarrow p$ where $p$ expresses the proposition that $b$ and $c$ have the same truth value. You take it from here.
Proposition a is false, if the two propositions b and c have different truth values; otherwise a is true.
Here is your toolbox.
$\neg x$ says "$x$ is false".
$x\wedge y$ says "$x$ is true and $y$ is true"
$x\vee y$ says "$x$ is true or $y$ is true (inclusive)"
$x\to y$ says "$y$ is true if $x$ is true"
$x\leftrightarrow y$ says "$y$ is true if $x$ is true else $y$ is false (so is $x$)" (PS: That is also "$x$ and $y$ have the same truth value") This may be expressed equivalently as $(x\to y)\wedge (\neg x\to\neg y)$ or $(x\wedge y)\vee(\neg x\wedge \neg y)$ , should you wish to avoid biconditionals.
So $a\vee (b\wedge c)$ says "$a$ is true or both $b$ and $c$ are true". That is not what you want. Try again.
