Sure! Here are the definitions for each of the terms you've requested:
Compound Inequality
A compound inequality is an inequality that combines two or more inequalities. It typically involves expressions that are related by the logical connectors "and" or "or." For example:
- And (conjunction): A compound inequality of the form \( a < x < b \) means that \( x \) lies between \( a \) and \( b \).
- Or (disjunction): A compound inequality such as \( x < a \) or \( x > b \) describes solutions for \( x \) that satisfy either one of the inequalities.
Empty Set
The empty set, often denoted by the symbol \( \emptyset \) or sometimes by curly braces \( { } \), is a set that contains no elements. It is the unique set that has a cardinality (size) of zero. In mathematical terms, the empty set is a subset of every set. The concept of the empty set is fundamental in set theory and serves as a basis for defining other sets.
Set Notation
Set notation is a mathematical way of describing a collection of objects (elements). There are several conventions for set notation, with some common forms being:
- Roster Form: Lists all the elements of the set explicitly. For example, \( A = {1, 2, 3} \).
- Set-builder Form: Defines a set by specifying a property that its members must satisfy. For example, \( B = { x \ | \ x \text{ is an even integer} } \).
- Interval Notation: Used to describe subsets of the real numbers. For example, \( [a, b] \) represents the closed interval of all real numbers between \( a \) and \( b \), inclusive, while \( (a, b) \) represents the open interval, excluding \( a \) and \( b \).
These forms of notation help succinctly convey information about sets and are widely used in mathematics.