Answer to Question #117101 in Discrete Mathematics for Priya

Question

Given the following statements as premises:
If he takes coffee, he does not drink milk.
He eats crackers only if he drinks milk.
He does not take soup unless he eats crackers.
At noon today, he had coffee.
Therefore he took soup at noon today.
Verify the validity of the statements.

Answer 1

Let

$p$: he takes coffe

$q$ : he drinks milk

$r$ :he eats crackers

$s$ : he takes soup

Then

1) If he takes coffee, he does not drink milk.

$p\to \bar{q}$

2) He eats crackers only if he drinks milk.

$r\to q$

3) He does not take soup unless he eats crackers.

$\bar{r}\to\bar{s}$

4) At noon today, he had coffee.

$p$

5) Therefore he took soup at noon today. 

$s$

Since

$p\to \bar{q}$ a premise

$\bar{q}\to\bar{r}$ contrapositive of premise

$p\to \bar{r}$ a conclusion by law of syllogism

$\bar{r}\to\bar{s}$ a premise

$p\to \bar{s}$ a premise law of syllogism

$p$ a premise

$\bar{s}$ a conclusion by modus ponens

Hence $\bar{s}$ is the conclusion

$(p\lor\bar{q})\to r$

$\begin{matrix} p & q&\bar{q}&r&p\lor \bar{q} &(p\lor\bar{q})\to r\\ F & F&T&F&T&F\\ F&F&T&T&T&T\\ F&T&F&F&F&T\\ F&T&F&T&F&T\\ T&F&T&F&T&F\\ T&F&T&T&T&T\\ T&T&F&F&T&F\\ T&T&F&T&T&T \end{matrix}$