Math Formula Vertical Bar

$Math Formula Vertical Bar - Formula Quest$

Math Formula: Understanding the Vertical Bar with Examples

The vertical bar symbol | plays a significant role in mathematics. Its versatility spans different branches of mathematics, including algebra, set theory, probability, and more. In this article, we will explore the uses of the vertical bar in various contexts, complete with examples to clarify its applications.

1. Vertical Bar in Absolute Value

The absolute value of a number represents its distance from zero on a number line. The formula is:

\[ |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases} \]

Examples:

$|5| = 5$
$|-3| = 3$
$|0| = 0$

2. Vertical Bar in Set Notation

In set theory, the vertical bar specifies conditions for elements in a set. It is read as "such that."

Example: $\{x | x > 0\}$ means the set of all $x$ such that $x > 0$.

Examples:

$\{n | n \text{ is even}, n \leq 10\} = \{2, 4, 6, 8, 10\}$
$\{y | y \in \mathbb{R}, y^2 = 4\} = \{-2, 2\}$

3. Vertical Bar in Probability

In probability, the vertical bar denotes conditional probability, defined as:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

Where:

$P(A | B):$ Probability of $A$ given $B$.
$P(A \cap B):$ Probability of both $A$ and $B$.
$P(B):$ Probability of $B$.

Example:

What is the probability of rolling an even number ($A$) given that the roll is greater than 2 ($B$)?

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{2}{4} = 0.5 \]

Conclusion

By understanding the uses of the vertical bar in different mathematical contexts, you can enhance your problem-solving skills. From absolute values to probability, this versatile symbol is indispensable in various fields of mathematics.