How many elements are there in the set ? mathematics question?

A set can be an element. Look at the top level, how many commas there are. {-1, {0}, 1, 2, {3,4}, 5} has 6 things separated by commas at the top level. It doesn't matter what those things are: Those things are -1, 1, 2, 5 and the two sets {0}, {3,4}. Those two sets are separated by commas from the other things, treat them the same way.

So that set has six elements. But what about the intersection you're being asked about? First let's look at that second set,
{z in Z : |z - 1| < 4}. Since this isn't in the form of a list, but of a rule ("the set of everything which satisfies this description") we have to do some reasoning about it. That says the set of integers 1, 2, 3, ... which are less than four units from 1.

Counting up, that includes 1, 2, 3, and 4 but not 5 (because |5 - 1| = 4 and the inequality is strictly <).
Counting down, that includes 0, -1, and -2 but not -3 (same reason, |-3 - 1| = 4).
So the second set is {-2, -1, 0, 1, 2, 3, 4}. The second set has 7 elements.

But you are being asked about the intersection. How many things appear in both lists? That automatically rules out those two sets that confused you in the first set, since there are no sets in the second set. What numbers appear in both lists? List them, count them.

The elements in the set on the right are -2, -1, 0, 1, 2, 3, 4
Of these only these three -1, 1, 2 are in the left set.

So answer is 3.