Question: What differences are there between the following properties:

> 
> $(\forall j \in \{1,2,3\}) ( \exists a \in \Bbb R) f\_j(a)=1$
> 
> 
> $ (\exists a \in \Bbb R) (\forall j \in \{1,2,3\}) f\_j(a)=1$
> 
> 
> $(\exists j \in \{1,2,3\}) ( \forall a \in \Bbb R) f\_j(a)=1$
> 
> 
> $ (\forall a \in \Bbb R) (\exists j \in \{1,2,3\}) f\_j(a)=1$
> 
> 
>

Answer: 
The first one says that for every $j$ in the set $\{1,2,3\}$, there exists an $a$ in the set of real numbers *dependent on $j$* such that $f\_j(a)=1$.

The second, that there exists an $a$ in the set of real numbers such that *for every* $j$ in the $\{1,2,3\}$, *no matter which one*, we have $f\_j(a)=1$.

Can you take it from here?

*Hint:*

> 
>  The third is like the first, except with $\{1,2,3\}$ and $\Bbb R$ swapped. The same is true of the fourth and second.
> 
> 
>
So you have three functions, named by the indices $\{1,2,3\}$, which accept real values as arguments (aka. their domain is points on the real number line).

> 
> $(\forall j \in \{1,2,3\}) ( \exists a \in \Bbb R)~~f\_j(a)=1$
> 
> 
> 

Each index has at least one argument which makes its function equal one. These arguments *need not* be the same.

Each function will equals one at some point. They need not do so at the same point.

> 
> $ (\exists a \in \Bbb R) (\forall j \in \{1,2,3\})~~f\_j(a)=1$
> 
> 
> 

There is an argument which makes the function of every index equal one. Such an argument must do so for all three functions.

At some point, all three functions do equal one simultaneously.

> 
> $(\exists j \in \{1,2,3\}) ( \forall a \in \Bbb R)~~f\_j(a)=1$
> 
> 
> 

For at least one index, its function is constantly $1$.

At least one function equals one everywhere.

> 
> $ (\forall a \in \Bbb R) (\exists j \in \{1,2,3\})~~f\_j(a)=1$
> 
> 
> 

Each real number argument will make *at least one* index's function equal one (not necessarily the same function at different points, but at least one of them at any).

At every point at least one of the three functions will equal one.