Help solving a predicate logic question:

Question

A(x,y,z): adding vector y to vector x results in vector z

Translate the statement into english: ∀x.∀y.∀z.A(x,y,z) --> A(y,x,z)

Answer 1

Why is this hard for you?   Where are you having trouble?   Take it step by step.

∀x.∀y.∀z.A(x,y,z) --> A(y,x,z)

Begin with the quantifiers.   $\forall$ become "any" and the domain is 'vectors', so:

Any vector x, any vector y, any vector z: A(x,y,z) --> A(y,x,z)

Move to the connectives.   It is a material implication; an "if-then" clause.

Any vector x, any vector y, any vector z: if A(x,y,z) , then A(y,x,z) .

Now for the predicates.   Ensure that you substitute the letters in the required order.

Any vector x, any vector y, any vector z: if adding vector y to vector x results in any vector z , then adding that vector x to that vector y results in that vector z.

Finally, rearrange to more natural language.

If adding any vector y to any vector x results in a vector z, then too adding vector x to vector y results in vector z.

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Well, that is as far as you need to go; it is what an exam marker would be looking for.   However you might want to take the extra step and eliminate the x,y,z symbols.   That is okay too.   Just make sure you preserve the intent and meaning of the statement.   ( On an exam I'd put both down, just to show you understood both the process, and meaning, of the statement. )

The order of addition of any two vectors is irrelevant, as either way yields the same result.

Which simply means that: "Vector addition is commutative."

Answer 2

For any vectors x,y, and z, if x+y=z then y+x=z.