Surjective

Definition

A function is called surjective (or onto) if every element in the codomain has at least one pre-image in the domain. This means that for every output value, there is at least one corresponding input value, ensuring that the function covers the entire codomain. Surjective functions are important because they help establish relationships between sets and are fundamental in understanding the behavior of composition and inverse functions.

5 Must Know Facts For Your Next Test

1. A surjective function guarantees that every element of the codomain is mapped to by at least one element of the domain, which means there are no 'unused' outputs.
2. If a function is surjective and has a finite domain, then the number of elements in the domain must be greater than or equal to the number of elements in the codomain.
3. Surjective functions can have multiple elements in the domain mapping to the same element in the codomain, leading to overlapping outputs.
4. In terms of inverse functions, only surjective functions can have a right inverse, which allows for a function to be inverted back to its original input.
5. The composition of two functions can be surjective if at least one of them is surjective; this property plays a key role in understanding how functions interact.

Review Questions

• How does the concept of surjectivity influence the existence of inverse functions?
  • Surjectivity directly impacts the existence of inverse functions because only surjective functions can have right inverses. A function must map every element in its codomain to ensure that when we attempt to reverse its mapping, we can reach every original input. If a function is not surjective, some elements in its codomain would lack pre-images, making it impossible to define an inverse for those missing mappings.
• Consider two functions, f and g. If g is surjective and f is any function, what can you say about the composition g ∘ f? Explain your reasoning.
  • If g is surjective, it ensures that every element in its codomain is mapped from some element in its domain. The composition g ∘ f will be surjective if f does not restrict the outputs too much. Specifically, as long as f maps enough elements from its domain into g’s domain so that g can cover its entire codomain, then g ∘ f will also be surjective. Otherwise, if f fails to map adequately, we might lose some outputs needed for surjectivity in the composition.
• Evaluate how understanding surjectivity can affect your approach to proving properties about functions within mathematical logic.
  • Understanding surjectivity allows you to create more robust proofs regarding properties of functions by establishing how inputs relate to outputs comprehensively. When demonstrating whether a function has certain characteristics or behavior, you can leverage surjectivity to show completeness in mapping and coverage. This conceptual framework aids in solving problems involving function interactions or proving whether compositions maintain properties like injectivity or bijectivity by considering how each part of a composition behaves with respect to coverage over their respective domains and codomains.

Related terms

Injective:

A function is injective (or one-to-one) if it maps distinct elements in the domain to distinct elements in the codomain, meaning no two different inputs share the same output.

Bijective: A function is bijective if it is both injective and surjective, meaning there is a perfect pairing between elements of the domain and codomain with no repeats.

Codomain: The codomain of a function is the set of all possible outputs that the function could theoretically produce, regardless of whether all values are actually achieved.