What is a null set?

I am very confused with null sets. I get that a set which has no elements will be called a null set but I am not getting the examples given below.

Please help me by explaining how $P,Q,R$ are all the null set? Thank-you

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Can you please clarify further? The page you linked to at Google books isn't available to everyone (it certainly isn't to me at least, so it's probably the same for someone else) –  kahen Jun 13 '11 at 13:52
@kahen: I have used the image from book now. –  Fahad Uddin Jun 13 '11 at 14:02
That's pretty awful writing to say "The null set... is denoted by $\phi$" and then four lines down redefine it with "The set $\phi$ = {0} is not a null set". The rest of the writing isn't that good either; I would find a different textbook if I were you. –  Rahul Jun 13 '11 at 16:24
@Rahul, @fahad: I agree with Rahul! If this is the text you're required to use for a class, perhaps you can supplement that text with another? I'm sure there are many good suggestions available here, depending on what level you're at in your studies. –  amWhy Jun 13 '11 at 23:16
@Doug: What would you make of a book all of whose variables, sets, functions, etc. were denoted "$\phi$" and no comment was made when switching between different meanings? No one was arguing with the statement that "$\{0\}$ is not a null set", the issue is with the unnecessary and uncommented-upon reuse of the symbol $\phi$. –  Zev Chonoles Jun 21 '11 at 2:57
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3 Answers

Perhaps what you find confusing is the use of set-builder notation to define $P, Q, R$: Included in between { ... } are the condition(s) that any "candidate" element must satisfy in order to be included in the set, and a set defined by set-builder notation contains all, and only, those elements satisfying all the conditions given.

In each of $P,\; Q, \;R$, set-builder notation is used to provide the conditions for inclusion in each set, respectively. Note: unless otherwise stipulated, you can take conditions separated by a comma to be a conjunction of conditions; that is: $$X = \{x : \text{(condition 1), (condition 2), ...., (condition n)}\}$$ means $X$ is the set of all x such that x satisfies (condition 1) AND x satisfies (condition 2) AND ... AND x satisfies (condition n).

$$P = \{x: x^2 = 4, x \text{ is odd}\}$$

The only solution to $x^2 = 4$ are $x = -2$ or $x = 2$, neither of which is odd. Hence there are $no$ elements in $P$; that is, $\;P = \varnothing$.

$$Q= \{x: x^2 = 9, x \text{ is even}\}$$

The only solutions to $x^2 = 9$ are $x = -3$ or $x = 3$, neither of which is even. Hence, there are no elements in $Q$; that is, $\;Q = \varnothing$.

$$R = \{x: x^2 = 9, 2x =4\}$$

$x = 2$ is the only solution to $2x = 4$, but $x = 2$ is not a solution to $x^2 = 9$, (and neither $x = 3$ nor $x = -3$ is a solution to $2x = 4$). Hence, there are no elements in $R$; that is, $\;R = \varnothing$.

NOTE: As an aside, regarding notation - sometimes instead of a colon :preceding the defining characteristics of a given element, you'll see | in place of the colon. E.g., $$P = \{x: x^2 = 4, x \text{ is odd}\}\iff \{x\mid x^2 = 4, x \text{ is odd}\}$$

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Thanks. Very nicely explained. It means that I can also make up my empty sets myself excluding those examples like {x:x=4 and x is a prime} like this? –  Fahad Uddin Jun 13 '11 at 14:58
@fahad: Yup...exactly! –  amWhy Jun 13 '11 at 15:04
Note that these are different ways to write down the empty set. There is only one due to extensionality. –  starblue Jun 13 '11 at 18:06
@starblue: yes, absolutely you are right, there is one empty set, which can be denoted in many ways... Similarly, there is one set of integers (the set of integers), which can be denoted in many ways... –  amWhy Jun 13 '11 at 18:46
I can't say that conditions separated by a comma qualify as a conjunction of conditions. If we have a conjunction of conditions, then such a conjunction qualifies as one condition. But, this doesn't come as necessary here. Where we have n conditions, each condition has to hold simultaneously. For example with {x:x^2=9, x is even} we don't have the single condition "conjunction of "x^2=9" and "x is even"," but rather that each of these conditions can get satisfied at the same time. That said, overall, nice response! –  Doug Spoonwood Jun 14 '11 at 14:53
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A Null Set is a set with no elements. While the author of your book uses the notation $\emptyset$, I prefer to use $\{\},$ to emphasize, that the set contains nothing. The example sets $P,\ Q$ and $R$ are all null sets, because there is no $x$, that can satisfy the condition of being included in the set.

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All sets have a null set as a subset of the set.

so: H={1,2} would have subsets of {1},{2},{1,2}, & {}.

The way I look at it is a set with literally nothing in it. So anything, even a 0, would not be a part of a null set.

So like amWhy and FUZxxl have said, P, Q & R are null sets because nothing that has been difined in the original example could ever have anything in them, so it would be null.

A more laymen example: If you have 5 apples and you define a subset of those apples to include all oranges that you have, it would be null because you have no oranges to put in the subset. (this is how it was explained to us in my discrete math class, lol)

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protected by Alexander Gruber♦Jun 15 '13 at 1:26

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