propositional calculus? I'm very stuck on this question in my High School class.
Atomic Sentances:
I – I am hungry
M – I will eat pie
V - I will become lazy.
B - I will be happy.
Hypothesis:
H1 – $I \implies M \land V$
H2 – $M \implies B$
H3 – $V \implies \neg B$
$I \implies M \land V, M \implies B , V ⇒ \neg B$
Show that 'I am not hungry' follows from the Hypothesis
I think I have to use Proof by contradiction for this but it is really confusing me? can anyone help please?
 A: We certainly do obtain a contradiction if we also know that $I$ holds. But if "I am not hungry", then "I do not eat pie" does not necessarily follow. So the conclusion necessarily follows if and only if $I$ holds.
If "I" is false, "V" is false, "B" is true, and M is true, all the premises still are true, and since "M" is true under this assignment, we have a consistent conclusion: "I eat the pie."
So the desired conclusion holds provided "I" is true.
UPDATE, given edit:
Okay, we can assume that $M \land V$. Then use your hypotheses to derive $B$ and $\lnot B$. Contradiction. Therefore, $\lnot (M \land V)$. Therefore, by modus tollens, $\lnot I$.
$(1) \quad I \implies M \land V\quad \text{hypothesis}$
$(2)\quad M \implies B\quad\quad\text{hypothesis}$
$(3)\quad V \implies \neg B\quad \text{hypothesis}$
$(4)\quad \quad\quad  \underline{M\land V\quad \quad\text{Assumption}}$
$(5)\quad\quad\quad \mid M\quad\quad\quad\text{$(4)\;\land$-elimination}$
$(6)\quad\quad\quad \mid V\quad\quad\quad\text{$(4)\;\land$-elimination}$
$(7)\quad\quad\quad \mid B\quad\quad\text{$(2, 5)\;$modus ponens}$
$(8)\quad\quad\quad \mid \lnot B\quad\quad\text{$(3, 6)\;$modus ponens}$
$(9) \quad\quad\quad\mid \text B \land \lnot B \quad\text{$(7, 8)\;\land$-introduction}$
$(10)\quad\quad\;\; \mid\;\;\perp\quad \quad \text{$(9)$ Contradiction}$
$(11)\quad \lnot(M \land V) \quad\text{$4 - 10\;\;\lnot$-Introduction}$
$(12)\quad \lnot I \quad\text{$(1)\;$ modus tollens}$
A: 1._$I$. (assumption) 
2.|$M\land V$ (by the first hypothesis and 1)
3.|$M$ (deduce from 2)
4.|$V$ (deduce from 2)
5.|$B$ (by the second hypothesis and 3)
6.|$\neg B$ (by the third hypothesis and 4)
7.|$B\land \neg B$ (from 5 and 6)
8.|contradiction
9.$I$
A: 'I am not hungry' shows that we want to get $\neg I$. We recall that the converse negative proposition equals the original proposition. So:
from H1: we have $\neg I\Leftarrow\neg M\vee\neg V\quad (1)$;
from H2: we have $\neg M\Leftarrow\neg B\quad (2)$;
from H3: we have $V\Rightarrow\neg B\quad (3)$ and $B\Rightarrow\neg V \quad (4)$.
With (2) and (3), we have $V\Rightarrow\neg M\quad(5)$, then with (4) and (1) we have:
$$V\vee B\Rightarrow\neg I$$
which means "if 'I become lazy' or 'I am happy', then 'I am not hungry'".
That's all!
