Sets, Relations and Functions Sets A set is a collection of distinct objects. A set can be represented by a symbol, such as {1, 2, 3, 4}. A set can have...
Sets, Relations and Functions
Sets
A set is a collection of distinct objects. A set can be represented by a symbol, such as {1, 2, 3, 4}. A set can have multiple elements, and it can be partitioned into distinct subsets.
Relations
A relation is a binary relationship between two sets. A relation can be represented by a table, where each row represents an element from the first set and each column represents an element from the second set. For example, the relation "is taller than" is a binary relation on the sets of people.
Functions
A function is a special type of relation where each element in the first set (domain) corresponds to exactly one element in the second set (range). For example, the function "area of a circle" is a function, where the domain is the set of all real numbers and the range is the set of all positive real numbers.
Equivalent Sets
Two sets are equivalent if they have the same elements. For example, the sets {1, 2, 3, 4} and {4, 5, 6, 7} are equivalent sets.
Symmetric Relation
A symmetric relation is a relation that is symmetric, meaning that for all elements a and b in the set, if a relates to b, then b relates to a. For example, the relation "is equal to" is a symmetric relation.
Transitive Relation
A transitive relation is a relation that is transitive, meaning that for all elements a, b, and c in the set, if a relates to b and b relates to c, then a relates to c. For example, the relation "is greater than" is a transitive relation.
Examples
The set {1, 2, 3, 4} is a set.
The relation "is taller than" is a binary relation on the sets of people.
The function "area of a circle" is a function.
The set {1, 2, 3} is equivalent to the set {3, 4, 5}.
The relation "is equal to" is a symmetric relation.
The relation "is greater than" is a transitive relation