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I have to prove whether or not these statements are true/false but I'm having trouble understanding it.

$\forall x \in \mathbb{Z}^5, \forall y \in \mathbb{Z}^5. \exists z \in \mathbb{Z}^5, \forall j \in \{1,2,3,4,5\},x_{j} \leq z_{j} \leq y_{j}$.

I think I understand the first half, up until the "for all j that is a member of the set {1,2,3,4,5}" part. Is it saying that that the element at index j of x will always be less than the one at z which is less than the one at y? How would you prove something like that (if it's true)?

The other one is this,

$i \neq j \implies (\{x \in \mathbb{Z}^5 : x_{i} = 3\} \cap \{x \in \mathbb{Z}^5 : x_{j} = 3\} = \emptyset)$

I'm not sure I understand what this one is saying at all. Maybe that the elements in the set of length 5 have to be at different indexes? But I'm not sure I understand why $x_{i} =3$.

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    $\forall x \in \mathbb{Z}^5$ so lets choose $x$ to be $(1,2,3,4,5)$ & $\forall y \in \mathbb{Z}^5$ lets choose $y$ to be $(5,4,3,2,1)$ ... now we need to find $z \in \mathbb{Z}^5$ so that $x_i \leq z_i \leq y_i$ for $i=1,2,3,4 & 5$ ... I don't think we will find such a $z$ for the fifth component ... So I guess it is false. – 2017-02-25

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Your interpretation of the first statement is more or less correct, but the statement itself is not.

The first statement claims that, for any pair of integer quintuples $x= (x_1...,x_5)$ and $y=(y_1,...,y_5)$ in the set $\mathbb{Z}^5$ there exists another integer quintuple $z = (z_1,...,z_5) \in \mathbb{Z}^5$ such that $x_i \leqslant z_i \leqslant y_i$ for each index $i=1,...,5$.

To see why this is false, take $x = (1,0,0,0,0)$ and $y=(0,0,0,0,0)$ as a counterexample. They are a legitimate pair of points in $\mathbb{Z}^5$, but there is no $z = (z_1,...,z_5) \in \mathbb{Z}^5$ satisfying $1 = x_1 \leqslant z_1 \leqslant y_1 = 0$.

The second statement says that the sets $A_i=\{(x_1,...,x_5) \in \mathbb{Z}^5: x_i = 3\}$ and $A_j=\{(x_1,...,x_5) \in \mathbb{Z}^5: x_j = 3\}$ are disjoint whenever $i \neq j$ are distinct indices among $1,...,5$. The set $A_i$ is the set of all integer quintuples whose $i$-th coordinate is $3$. Similarly, the set $A_j$ is the set of all integer quintuples whose $j$-th coordinate is $3$. So, the statement essentially claims that an integer quintuple cannot have distinct coordinates both equal to $3$.

This certainly isn't true. The perfectly legitimate quintuple $x = (3,3,3,3,3) \in A_i \cap A_j$ serves as a counterexample

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    That makes so much more sense, thanks! – 2017-02-25
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    No problem. The notation of quantifiers is indispensable, but if ever unclear, spelling things out in full sentences tends to help me. – 2017-02-25