What do the events A+A and AA represent?
What do the events $A+A$ and $AA$ represent?
Explanation of Method
$A+A$ is the union of $A$ with itself and $AA$ is the intersection of $A$ with itself. We check, outcome by outcome, which outcomes belong to each event. Saying the same condition twice ("$A$ or $A$", "$A$ and $A$") adds no new information, so both events turn out to be just $A$. These are the idempotent laws. We also point out how this differs from ordinary arithmetic with numbers.
Step 1: The event $A+A$
By definition $A+A = A\cup A$ is the event "$A$ occurs or $A$ occurs". For any outcome $\omega$:
\[ \omega\in A+A \iff \omega\in A \ \text{ or } \ \omega\in A \iff \omega\in A. \]
The statement "P or P" is true exactly when P is true, so the two events contain the same outcomes:
\[ \boxed{A + A = A}. \]
In words: "$A$ occurs or $A$ occurs" says nothing more than "$A$ occurs".
Step 2: The event $AA$
By definition $AA = A\cap A$ is the event "$A$ occurs and $A$ occurs". For any outcome $\omega$:
\[ \omega\in AA \iff \omega\in A \ \text{ and } \ \omega\in A \iff \omega\in A. \]
The statement "P and P" is true exactly when P is true, so
\[ \boxed{AA = A}. \]
In words: "$A$ occurs and $A$ occurs" is the same as "$A$ occurs".
Step 3: Picture it with a Venn diagram
Shade the circle $A$. For $A+A$ we shade "everything in $A$ or in $A$" — the same circle. For $AA$ we shade "the common part of $A$ and $A$" — again the same circle. Combining a region with itself never changes it.
Step 4: Example with a die
Roll a die, $\Omega = \{1,2,3,4,5,6\}$, and let $A$ = "an even number" $= \{2,4,6\}$.
\[ A + A = \{2,4,6\}\cup\{2,4,6\} = \{2,4,6\} = A, \qquad AA = \{2,4,6\}\cap\{2,4,6\} = \{2,4,6\} = A. \]
Step 5: Why these are called idempotent laws
"Idempotent" comes from Latin idem ("the same") and potens ("power"): applying the operation to an object and itself gives back the same object. More generally, for any number of repetitions,
\[ A + A + \dots + A = A, \qquad A\cdot A\cdots A = A. \]
Step 6: Common mistakes to avoid
- Events are not numbers. For a number $a$ we have $a+a = 2a$ and $a\cdot a = a^2$. For events there is no "$2A$" and no "$A^2$": $A+A$ and $AA$ are simply $A$.
- $AA$ does not mean "$A$ happens twice". All events in a product refer to the same trial. If a die is rolled twice and we want "an even number both times", we need two different events: $A_1$ = "even on the first roll", $A_2$ = "even on the second roll", and the event is $A_1A_2$. Its probability is $P(A_1)P(A_2) = \tfrac12\cdot\tfrac12 = \tfrac14$, while $P(AA) = P(A) = \tfrac12$.
Result
| Event | Result | Meaning | Name |
|---|---|---|---|
| $A+A$ | $A$ | "$A$ or $A$" is the same as "$A$" | idempotent law for the sum (union) |
| $AA$ | $A$ | "$A$ and $A$" is the same as "$A$" | idempotent law for the product (intersection) |