Propositional Logic Inductive Proof I am working on a problem to prove, but I do not understand it completely. Where should I use inductive method? What is the base case? And so on. Here is my problem:

A truth assignment $M$ is a function that maps propositional variables
  to $\{0, 1\}$ ($1$ for true and $0$ for false). We write $M\vDash x$ if $x$ is true
  under $M$. We define a partial order $\leq$ on truth assignments by 
  $M \le M'$ if $M(p) \le M'(p)$ for every propositional variable $p$.  
A propositional formula is positive if it only contains connectives $\wedge$
  and $\vee$ (i.e., no negation $\lnot$ or implication $\Rightarrow$).  
Use Proof By Induction
  to show that for any truth assignments $M$ and $M'$ such that $M\le M'$, and
  any positive propositional formula $x$, if $M \vDash x$, then $M' \vDash x$.

I am really confused. Any help is welcome. Thank you.
 A: Perhaps a more concrete statement of what you need to show is the following:

Let $M \leq M^\prime$ be truth assignments.  Then for every positive formula $\varphi$ either $M \not\models \varphi$ or $M^\prime \models \varphi$.

We also note that the family of positive formulas has its own inductive definition, similar to the definition of the family of all formulas:  

  
*
  
*Every propositional variable $p$ is a positive formula.
  
*If $\varphi , \psi$ are positive formulas, then so are $\varphi \wedge \psi$ and $\varphi \vee \psi$.
  
*(No other formula is positive.)
  

So to prove the result, we need to show two-and-a-half things: Let $M \leq M^\prime$ be truth assignments, then


*

*if $p$ is a propositional variable, then either $M \not\models p$ or $M^\prime \models p$.

*if $\varphi , \psi$ are formulas such that either $M \not\models \varphi$ or $M^\prime \models \varphi$, and similarly for $\psi$, then

*

*either $M \not\models \varphi \wedge \psi$ or $M^\prime \models \varphi \wedge \psi$; and

*either $M \not\models \varphi \vee \psi$ or $M^\prime \models \varphi \vee \psi$.



To leave with a parting
Hint: The definition of the partial order $\leq$ will be very useful in the base case (1), and the definition of $\models$ (for compound formulas) will be very useful in the inductive step (2).
