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Write a truth table for the Boolean formula (X → Y )AND NOT Y . Include the intermediate calculations of (X →Y ) and NOT Y as in the incomplete example below:

X Y (X → Y ) NOT Y (X →Y ) ^ NOTY

tt......"T"......."F"......."F"

t f....."F"......."T"......."F"

f t....."T"......."F"......."T"

f f....."T"......."T"......."T"

Look at the lines of the truth table in which the outcome is true". What do you notice about the values of X on those lines? "Therefore not x EG x is not true by contradiction"

Now suppose I tell you that: _ if it is raining when I get up in the morning, I pack my raincoat; _ I did not pack my raincoat today. What can you conclude about the weather when I got up today? "It was not raining"

My answer attempt so far are in quotes above.

…am I on the right track?

Regards j

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  • $\begingroup$ My answer so far are in double quotes. $\endgroup$
    – user103388
    Oct 26, 2013 at 21:33

1 Answer 1

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You made one mistake; I’ve corrected it in red below.

$$\begin{array}{cc} X&Y&X\to Y&\neg Y&(X\to Y)\land\neg Y\\ \hline T&T&T&F&F\\ T&F&F&T&F\\ F&T&T&F&\color{red}F\\ F&F&T&T&T \end{array}$$

Your conclusion that if $(X\to Y)\land\neg Y$ is true, then $X$ must be false still holds, and in fact you can also conclude that $Y$ must be false: $(X\to Y)\land\neg Y$ is logically equivalent to $\neg X\land\neg Y$. And because your conclusion that $X$ must be false is correct, your answer to the final part of the question is also correct.

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