COMPLEMENT

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Complement

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◇Overview

In set-theoretic probability, the Complement of an event $A$ (denoted $A^c$, $A'$, or $\overline{A}$) is the set of all outcomes in the sample space $S$ that are not in $A$. Every outcome in $S$ belongs to either $A$ or $A^c$, never both.

◇Key Definitions

• Set Definition: $A^c = \{\omega \in S : \omega \notin A\} = S \setminus A$.
• Partition Property: $A \cup A^c = S$ (collectively exhaustive) and $A \cap A^c = \emptyset$ (mutually exclusive).
• Involution Law: $(A^c)^c = A$.
• Boundary Complements: $S^c = \emptyset$ and $\emptyset^c = S$.

◇Worked Example

Problem: Let the sample space of rolling a fair die be $S = \{1, 2, 3, 4, 5, 6\}$, and let $A = \{2, 3, 5\}$ be the event of rolling a prime number. Find $A^c$ and $P(A^c)$.

Solution:

• $A^c = S \setminus A = \{1, 4, 6\}$ (note that $1$ is not prime).
• $P(A^c) = \frac{|A^c|}{|S|} = \frac{3}{6} = 0.50$.

◇References & Navigation

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• Related Notes: ◇Union, ◇Counting Techniques