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

 A: 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., $$P = \{x: x^2 = 4, x \text{ is odd}\}\iff \{x\mid x^2 = 4, x \text{ is odd}\}$$
A: 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.
A: 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)
