Injectivity in discrete math refers to a function that maps distinct elements from the domain to distinct elements in the codomain. This means that each input in the domain has a unique output in the codomain, and no two inputs have the same output.
To prove that a function is injective, you must show that for every distinct pair of inputs in the domain, the corresponding outputs in the codomain are also distinct. This can be done using a direct proof or a proof by contradiction.
Proving injectivity is important because it ensures that a function has a well-defined inverse. This is useful for solving equations, calculating probabilities, and understanding the behavior of a function.
Yes, the function f(x) = x + 2 is injective because for every distinct input, the output is also distinct. For example, f(3) = 5 and f(4) = 6, showing that the function maps distinct inputs (3 and 4) to distinct outputs (5 and 6).
If you are unable to prove that a function is injective, it may not be injective. In this case, you can try to find a counterexample (a pair of inputs that have the same output) to disprove injectivity. If a counterexample is found, the function is not injective. Otherwise, you may need to use other methods, such as checking for surjectivity or bijectivity.