What is the difference between following Predicate Logic?

Question

$\forall x \forall y \left (( Mother(x) \ \Lambda\ Loves(x,y)\right) \rightarrow child(y,x))$

$\forall x \forall y\left (( Mother(x) \ \Lambda\ child(y,x)\right) \rightarrow Loves(x,y))$

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For 2nd one, I think it is appropriate to say

Every Mother loves her children

But, I am unable to translate 1st one into English ?

Answer 1

This statement is like the statement 'Only Americans are nice', which would be translated as:

$\forall x (Nice(x) \rightarrow American(x))$

but may be more easily understood by looking at its contrapositive equivalent:

$\forall x (\neg American(x) \rightarrow \neg Nice(x))$

We can do the same for this statement about mothers:

$\forall x \forall y\left (( Mother(x) \ \land\ Loves(x,y)\right) \rightarrow Child(y,x)) \Leftrightarrow$

$\forall x \forall y\left (Mother(x) \rightarrow (Loves(x,y)\right) \rightarrow Child(y,x))) \Leftrightarrow$

$\forall x \forall y\left (Mother(x) \rightarrow (\neg Child(y,x)\right) \rightarrow \neg Loves(x,y))) $

And so now you get that a mother does not love anything that is not their child or: ''Mothers only love their children"

By the way, in the first step I used Exportation, which says $P\rightarrow (Q \rightarrow R) \Leftrightarrow (P \land Q) \rightarrow R$

Answer 2

Literally you can see it is saying if a mother loves something, then that something is necessarily her child. I'll let you translate it more eloquently.