number systems?

Question

i am trying to understand number systems, i have this question:
Z is a subset of N (Z E N). [the E in the bracket is supposed to be a small curled e]. how would i know if this was true or false?
would someone please be able to explain?

Answer 1

ZEX20913 is basically right but I think his interpretation does not work.

If the expression is:

Z ∊ N

then it means that Z is an element of N. If N is the set of Natural numbers then its 'elements' are numbers, NOT sets.

If the expression is

Z ⊆ N

then it means that Z is a subset (but not necessarily a strict subset) of N.

However if Z is the set of Integers then Z ISN'T a subset of N because there are elements in Z (e.g. -1) which are not in N.

So if you have decide whether this is true, the method is to look for an element which is in Z but not in N, and that shows you the expression is false.

Remember that although the set of Integers and the set of Natural Numbers are both infinite one contains an infinite number of elements not contained in the other one (ie all the negative numbers present in the Integers but not in the Natural numbers).

Answer 2

Given two sets A and B, we say that A is a subset of B if and only if every element that is in A is also in B. Conversely, we say B is a subset of A if and only if every element that is in B is also in A.

Applying this to your problem, to know whether it is true or false, you need to know whether every element of Z is in N. This might sound difficult, given that Z is an infinite set, but notice that N does not contain any negative integers (i.e. N doesn't contain numbers like -1,-2,-3,...). However, -1,-2,-3,... are definitely whole numbers and hence are in the in Z.

Thus, as there are elements of Z that are not in N, we know that Z is not a subset of N. In fact, N is a subset of Z, since the set of natural numbers is the set of positive, or depending on who you ask, non-negative, numbers integers. That is, every natural number is an integer, and hence N is a subset of Z.

Hope this is helpful!

Answer 3

Z is the integers. N is the natural (or counting) numbers.

Do you know what these are?

An integer is a number with no decimal part.

1 is.
-45 is.
.5 is not.

A natural number is a positive integer (sometimes 0 is allowed in the club.)

The E represents the idea "Belongs to", but for this, it should be more like a c with an underline than an E. This C represents subset, and the E represents an Element of the set.

This should be more than enough to answer your question.

Answer 4

The natural numbers are positive integers greater than 0:
1,2,3,...

Z is the integers:
0, +1, -1, +2, -2, ...

So, N is a subset of Z, but not the converse.

To prove that falsety of the statement just show an exception:

Take x=-1, x is an element of Z but not N
therefore the statement is false.

Btw, Z comes from the german word for number.