properties of injective functions
Suppose that for some . By definition of composition, . Since , is assumed injective, . Since is also assumed injective, . Therefore, implies , so is injective. ∎
Suppose is an injection, and . Then the restriction is an injection.
Suppose for some . By definition of restriction, . Since is assumed injective this, in turn, implies that . Thus, is also injective. ∎
Suppose are sets and that the functions and are such that is injective. Then is injective.
(direct proof) Let be such that . Then . But as is injective, this implies that , hence is also injective. ∎
(proof by contradiction) Suppose that were not injective. Then there would exist such that but . Composing with , we would then have . However, since is assumed injective, this would imply that , which contradicts a previous statement. Hence must be injective. ∎
It follows from the definition of that , whether or not happens to be injective. Hence, all that need to be shown is that . Assume the contrary. Then there would exist such that . By defintion, means , so there exists such that . Since is injective, one would have , which is impossible because is supposed to belong to but is not supposed to belong to . ∎
Suppose is an injection. Then, for all , it is the case that .
Whether or not is injective, one has ; if belongs to both and , then will clearly belong to both and . Hence, all that needs to be shown is that . Let be an element of which belongs to both and . Then, there exists such that and such that . Since and is injective, , so , hence . ∎
|Title||properties of injective functions|
|Date of creation||2013-03-22 16:40:20|
|Last modified on||2013-03-22 16:40:20|
|Last modified by||rspuzio (6075)|