Question: What differences are there between the following properties:

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> $(\forall j \in \{1,2,3\}) ( \exists a \in \Bbb R) f\_j(a)=1$
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> $ (\exists a \in \Bbb R) (\forall j \in \{1,2,3\}) f\_j(a)=1$
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> $(\exists j \in \{1,2,3\}) ( \forall a \in \Bbb R) f\_j(a)=1$
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> $ (\forall a \in \Bbb R) (\exists j \in \{1,2,3\}) f\_j(a)=1$
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Answer: 
The first statement says: for each $j$ in $\{1,2,3\}$, you can find a real number $a$ (which number may depend on which $j$) with the property that the $j$th function maps $a$ to $1$. 

Presumably, you have three functions called $f\_1$, $f\_2$, and $f\_3$, and the first statement tells you that each and every one of these three functions "hits" the value $1$ at some point. It need not be the same point for every function.

The second statement says that there is a real number $a$ with the property that each of the three functions $f\_1$, $f\_2$, and $f\_3$ send $a$ to $1$; here, $a$ is *independent* of the function: the same number $a$ has to "work" for all three functions.

The third statement says that there is at least one of the three functions that sends *every* real number to $1$; that is, at least one of the functions is the constant function $1$.

The fourth statement says that if you take a real number $a$, you will be able to find that at least one of the functions $f\_1$, $f\_2$, and $f\_3$ send $a$ to $1$, and that this works for *every* real number $a$.

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Graphically: imagine you draw the graphs of the three functions on the same plane. The first statement says that the horizontal line $y=1$ will intersect all three graphs, at least once for each graph; but it may intersect them at different points for each graph.

The second statement says that there is a number $a$ such that all three graphs go through $(a,1)$: the vertical line $x=a$ touches all three graphs at height $1$. 

The third statement says that at least one of the graphs *is* the horizontal line $y=1$.

And the fourth statement says that every vertical line $x=a$ intersects at least one of the graphs at height $1$, but which graph it intersects may be different for different values of $a$.
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:*

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>  The third is like the first, except with $\{1,2,3\}$ and $\Bbb R$ swapped. The same is true of the fourth and second.
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