Strong monotonicity and weak monotonicity

Question

We say $\succsim$ represents weak monotonic preferences if $$x,y \in X, \,\, y >> x \implies y \succ x $$

where $y >> x$ means that every element of $y$ is greater than every element of $x$.

And we say $\succsim$ represents strong monotonic preferences if $$x,y \in X, \,\, y \geq x, y \neq x \implies y \succ x$$

where $y \geq x$ means that at least one element of $y$ is greater than an element of $x$ and all others are equal.

My question is: do strong monotonic preferences imply weak monotonic preferences?

My answer: yes. The reasoning is as follows:

Since the set $A = \{ x,y \in X: y >> x \}$ is a subset of $B = \{x,y \in X: y \geq x, y \neq x \}$, then if $B \implies y \succ x$ it must also be true that $A \implies y \succ x$ and thus strong monotonic preferences imply weak monotonic preferences.

In plain English, if a bundle with more of one commodity and the same of all others is preferred, then a bundle with more of every commodity must also be preferred.

Is my reasoning correct?

Thanks!

Answer 1

Let $x,y\in X$. Suppose $y \gg x$ (so that in particular, $y\geq x$ and $y\neq x$). By assumption, $\succsim$ satisfies strong monotonicity; and so, $y\succ x$.

(We've just shown that for any $x,y\in X$ such that $y \gg x$, we have $y\succ x$. Hence, $\succsim$ satisfies weak monotonicity.)

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The argument you've given above is correct and roughly the same as the proof I've just given. The only problem is that your argument is somewhat indirect, convoluted, and unclear.