To prove that a composition is a mapping, we need to show that for any two elements a and b in the domain of the composition, the composition f(g(a)) and f(g(b)) are well-defined and unique. This means that for any input a, there is a unique output f(g(a)). We also need to show that the composition is closed under the given operation, meaning that the composition of any two elements in the domain results in an element in the codomain. Finally, we need to show that the composition satisfies the properties of a mapping, such as associativity and identity. By demonstrating these properties, we can prove that the composition is indeed a mapping. **

A mapping is proportional when there is a constant ratio between the corresponding values of the two sets being mapped. In other words, if the ratio of the output values to the input values remains constant, then the mapping is considered proportional. This means that as one set of values increases or decreases, the other set of values changes in direct proportion. **

Yes, the mapping is left-unique. This means that each input value in the domain is associated with only one output value in the range. In other words, no two different input values can map to the same output value. This ensures that the mapping is well-defined and unambiguous. **

MIDI mapping allows users to assign specific MIDI messages to control parameters within a software or hardware device. This process involves selecting a parameter to be controlled, such as volume or effects, and then assigning a MIDI message, such as a note, controller, or program change, to that parameter. Once the mapping is set up, the MIDI message can then be sent from a MIDI controller, such as a keyboard or pad controller, to manipulate the assigned parameter in real-time. This allows for a more hands-on and customizable approach to controlling music software and hardware. **

An identical mapping is a type of mapping where each element in one set is paired with a unique element in another set, such that the pairing preserves the identity of the elements. In other words, each element in the first set corresponds to only one element in the second set, and vice versa. This type of mapping is often used in mathematics and computer science to establish a one-to-one correspondence between elements of two sets. **

A linear mapping, also known as a linear transformation, is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, it maps a vector from one space to another in a way that maintains the structure of the vector space. Linear mappings are fundamental in linear algebra and are used to describe various mathematical concepts and relationships in a geometrically meaningful way. They play a crucial role in solving systems of linear equations, studying eigenvalues and eigenvectors, and understanding the properties of matrices. **

A bijective mapping is a function between two sets that is both injective and surjective. In other words, every element in the domain is paired with a unique element in the codomain, and every element in the codomain is paired with at least one element in the domain. This means that there is a one-to-one correspondence between the elements of the two sets. Bijective mappings are also known as one-to-one and onto functions. **

One can represent a mapping by using a diagram or a table. In a diagram, the mapping is typically shown with arrows connecting the input values to their corresponding output values. In a table, the mapping can be represented by listing the input values in one column and their corresponding output values in another column. Additionally, mathematical notation can also be used to represent a mapping, such as using function notation (e.g., f(x) = y). **