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Suppose "If $A$, then $B$ and not $C$" is true. Is the following statement true?

If $A$ and $C$, then not $B$.

I know the answer is true but I don't know the basis behind it.

Bart Michels
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user127340
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2 Answers2

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$X \implies Y$ is true if $X$ is false or $X$ and $Y$ are true.

Thus, if $A$ is false, the first statement is true. Also the second statement is true if $A$ is false for the same reason.

On the other hand if $A$ is true, then $C$ has to be false (and $B$ has to be true but that does not matter) in the first statement. But if $C$ is false the second statement is true!

Hoewever, you could also do a truth table to prove this assumption $$\begin{array}{c|c|c|c|c} A&B&C&A\implies(B\wedge\lnot C) & (A\wedge C)\implies\lnot B\\ \hline 0&0&0&\color{blue}{1}&\color{blue}{1}\\ 0&0&1&\color{blue}{1}&\color{blue}{1}\\ 0&1&0&\color{blue}{1}&\color{blue}{1}\\ 0&1&1&\color{blue}{1}&\color{blue}{1}\\ \hline 1&0&0&0&1\\ 1&0&1&0&1\\ 1&1&0&\color{blue}{1}&\color{blue}{1}\\ 1&1&1&0&0\\ \end{array}$$

We can see here that both statements are not equivalent but everytime the first statement is true, the second statement is true as well

user127.0.0.1
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You can forget about B entirely. Since A implies $\lnot$C, the compound statement "A and C" is false. Since the hypothesis of your second conditional is false, the conditional statement as a whole is true no matter what you write as the conclusion.

Austin Mohr
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  • I understand what you're saying but how do you know that A and C is false from A implies not C? – user127340 Feb 09 '14 at 06:48
  • @user127340 A is either true or false. If it is false, then A$\wedge$C is false. If it is true, then it implies C is false, which in turn implies A$\wedge$C is false. – Austin Mohr Feb 09 '14 at 07:06