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Sets and functions are foundational concepts in mathematics, particularly in the field of algebra and analysis. Here’s a brief overview:

1. **Sets**: A set is a collection of distinct objects, considered as an object in its own right. Sets are usually denoted by curly braces `{}`. For example, the set of all even numbers less than 10 can be written as `{2, 4, 6, 8}`. Some important terms related to sets include:
– Elements: Objects that belong to a set. In the set `{1, 2, 3}`, the numbers 1, 2, and 3 are elements of the set.
– Subset: A set A is a subset of another set B if every element of A is also an element of B. The symbol for subset is ⊆. For example, `{1, 2}` is a subset of `{1, 2, 3}`.
– Universal set: The set that contains all the objects under consideration. It is often denoted by the symbol Ω.
– Empty set: The set with no elements, denoted by {} or ∅.

2. **Functions**: A function is a relation between a set of inputs (the domain) and a set of possible outputs (the codomain), where each input is related to exactly one output. Functions are often denoted by a rule that describes how the input is transformed into the output. For example, the function f(x) = x^2 squares its input. Some key concepts related to functions include:
– Domain: The set of all possible inputs (x-values) for which the function is defined.
– Codomain: The set of all possible outputs (y-values) that the function can produce.
– Range: The set of all actual outputs that the function produces for the inputs in the domain. The range is a subset of the codomain.
– One-to-one (Injective) function: A function in which every element of the domain maps to a unique element in the codomain.
– Onto (Surjective) function: A function in which every element in the codomain is mapped to by at least one element in the domain.
– One-to-one correspondence (Bijective function): A function that is both injective and surjective, meaning it is a one-to-one mapping between the domain and codomain.

Understanding sets and functions is crucial in many areas of mathematics, including calculus, algebra, and discrete mathematics.