Use of "for all" in definition of reflexive and symmetric relations.

Question

My book says that a relation R on A is

1. reflexive, if $\ (a,a) \in R, \ for\ every \ a \in A$
2. symmetric, if $\ (a_1,a_2) \in R \implies (a_2,a_1) \in R,\ for\ all\ a_1,a_2 \in A$

Although I know what a symmetric relation is, I can't understand the formal definition. My interpretation of definition 2 is that because of the "for all", all the ordered pairs should be a part of R, in other words $R = A \times A $ which is obviously wrong.

So, my question is that why is the "for all" present in the 2nd definition? Also, do "for all" and "for every" suggest different things in the definition?

Answer 1

The statement should be read as:

For all $a_1, a_2 \in A$: If $(a_1, a_2) \in R \Rightarrow (a_2, a_1) \in R$

Hence we don't consider all pairs $(a_1, a_2) \in A$, only $(a_1, a_2) \in R$. However, every element $a_i \in A$ may be used to "try" to form these pairs.

Answer 2

The meaning of quantifiers can be confusing sometimes. A clearer way to phrase these definitions would be to put the "for all" expression before the conclusion. The new definitions would look like this:

1. A relation $R$ is reflexive if for all $a\in A$, $(a,a)\in R$.
2. A relation $R$ is symmetric if for all $a_1,a_2\in A$, $(a_1,a_2)\in R\implies(a_2,a_1)\in R$.

As to your second question, "for any", "for every", and "for all" mean the same thing in mathematics, although the clearest is probably "for all".