The distinction between the empty set and the number is similar to that between NULL and ZERO. For example, the set of real solutions (or informally "the solution") to is , but the solution to is .

From MSDN:

The predefined unary and binary operators and any user-defined operators that exist for value types may also be used by nullable types. These operators produce a null value if the operands are null; otherwise, the operator uses the contained value to calculate the result.

That's why all the test are passed, including the last one - no matter what the operand value is, if another operand is null, then the result is null.

Null: A value of NULL indicates that the value is unknown. A value of NULL is different from an empty or zero value. No two null values are equal. Comparisons between two null values, or between a NULL and any other value, return unknown because the value of each NULL is unknown.

If you do any arithmetic operations with null the whole expression evaluates to Null. In order to handle null you should use Isnull() or coalesce function like this.

select 1 - isnull(NULL,0) as result

This is sufficiently rare, and handled in sufficiently many different ways, that you should always state explicitly how you're treating it. In my experience, the most common symbols are:

• $\mathsf{null}$, $\mathsf{nil}$, $\mathsf{NaN}$ or similar on the more computer-sciencey side, and

• $\perp$ or $\uparrow$ on the more logicy side.

  • Note that "$\perp$" is also used to denote contradiction, and "$\uparrow$" is also used as a predicate to denote "is undefined" or "doesn't halt" with "$\downarrow$" denoting "is defined"/"does halt."

But again, I'd explicitly state which you're using - although admittedly multiple of these would almost certainly make it obvious from context.

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

NULL (as far as my interpretation goes) is unrepresentable data. The only appropriate tests for null are IS NULL, IS NOT NULL, and several functions made specifically to handle NULL values: http://dev.mysql.com/doc/refman/5.0/en/comparison-operators.html

You could say - IFNULL(col_c, 0)+col_d (COALESCE will work identically in this case).

More information on working with NULLs: http://dev.mysql.com/doc/refman/5.0/en/problems-with-null.html