Taking a hypothesis and using a list of tautologies to prove a conclusion

Question

So I had this problem on an old homework that I didn't really understand.

In each part a list of hypotheses are given. These hypotheses are assumed to be true. Using tautologies, you are to establish a desired conclusion. Indicate which tautology you are using to justify each step.

Hypothesis: r $\Rightarrow$ $\lnot$s , $\lnot$r $\Rightarrow$ $\lnot$t , $\lnot$t $\Rightarrow$ u, v$\Rightarrow$s

Conclusion: $\lnot$v $\lor$ u

So I went to office hours for this question and my professor pretty much reiterated the hint section in the back of hour textbook. What I am stuck on is how to use all these hypotheses to prove this conclusion. Can anybody guide me on how to approach a problem like this?

here is a list of tautologies for reference: http://www.math.ucsd.edu/~jeggers/math109/tautologies.pdf

Answer 1

One approach is to see what you feel you can deduce from the given hypotheses, and then separately concentrate on how to express that deduction using tautologies. For example: $r$ implies $\lnot s$, and $\lnot s$ implies $\lnot v$ (by the contrapositive of $v\implies s$; so if $r$ is true then $\lnot v$ is true. Similarly, what can you deduce if $\lnot r$ is true? And then, one of $r$ and $\lnot r$ has to be true....

Answer 2

\begin{cases}v\implies s&(1)\text{ Given}\\r\implies\neg s&(2)\text{ Given}\\s\implies\neg r&(3)\text{ Contrapositive of }(2)\\v\implies\neg r&(4)\text{ Hypothetical Syllogism }(1),(3)\\\neg r\implies\neg t&(5)\text{ Given}\\v\implies\neg t&(6)\text{ Hypothetical Syllogism }(4),(5)\\\neg t\implies u&(7)\text{ Given}\\v\implies u&(8)\text{ Hypothetical Syllogism }(6),(7)\\\neg v\vee u&\text{Implication }(8)\end{cases}