discrete math, distinction between and and implies, exemplified

Question

Please refer to the problem below. I have difficulty distinguishing when to put a 'and' and when to put an 'implied' in these 2 questions.(As circled in red).

Answer 1

The word where has a very different meaning in the two sentences. It's a good example of why we use mathematical notation to avoid confusion.

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In the second sentence, you can replace the word where with the word if, and you get a sentence

If $m>n$, $n^2+m>10$

And you should know that a sentence that looks like "If $X$, then $Y$" can simply be written as "$X\implies Y$".

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In the first example, you cannot make the same switch, i.e. you cannot replace "where" with "if", so the statement $q\neq r$ stands on its own as a separate statement.

Answer 2

Get rid of the where expressions. For the first statement this is quite easy, because you can rephrase it as follows without changing its meaning:

There are non-negative integers $q$ and $r$ such that $q\ne r$ and $q+r=5$.

When you phrase it this way, it’s pretty clear that you’re asserting the existence of non-negative integers $q$ and $r$ that satisfy both the condition $q\ne r$ and the condition $q+r=5$:

$$\exists q\in\Bbb N\,\exists r\in\Bbb N\,(q\ne r\land q+r=5)\;.$$

The second is a little trickier. One way is to notice that where $n>m$ just limits which positive integers $n$ and non-negative integers $m$ you’re really talking about. If $n=2$ and $m=3$, for instance, nothing is actually being asserted, because $n\not>3$. It’s really a statement about those $n$ and $m$ such that $n>m$:

For each positive integer $n$ and non-negative integer $m$ such that $n>m$, it is true that $n^2+m>10$.

If we’re willing to be a bit more wordy, we can expand this as follows:

For each positive integer $n$ and non-negative integer $m$, if $n>m$ then $n^2+m>10$, and if $n\le m$ we make no claim at all.

The last clause of this version doesn’t really add anything: we can omit it without changing the meaning. That is, instead of explicitly stating that we make no claim when $n\not>m$, we simply don’t make any claim in that case:

For each positive integer $n$ and non-negative integer $m$, if $n>m$ then $n^2+m>10$.

This version pretty clearly translates into symbols as

$$\forall n\in\Bbb Z^+\,\forall m\in\Bbb N\,(n>m\to n^2+m>10)\;.$$