Topic 1.4 Functions - Injections, Surjections and Bijections Flashcards
There’s some information here, can you remember the information I mean?
Everything in the codomain is an image of something in the domain. The range and the codomain are part of the same set, where the range is always part of the codomain, but the range could be smaller because maybe not everything in the codomain is an output.
What is the set ‘W’?
It is the set of all English words
What is the set ‘A’?
The roman alphabet
What does it mean when we say that ‘f’ is injective?
Injectivity means the function preserves distinctness. It means that if the inputs are different, then the outputs must also be different (if a1≠a2, then f(a1)≠f(a2) for every pair a1,a2∈A)
Can you influence the injectivity of something?
In injectivity of a function can normally be influenced by changing the domain, since generally if you are hoping to ensure there is only one output per input, and the codomain is generally a far larger set, it would be easier to simply modify the domain (recall Horizontal Line Test, but just know it isn’t the best calc. 1 method)
How does monotone increasing/decreasing influence definitions of injectivity?
You can tell if ‘f’ is injective by whether or not it is either always monotone increasing or always monotone decreasing
What does it mean when we say ‘f’ is monotone increasing?
It means that if whenever you increase the input, the output must also increase (uphill slope). Written as: for every pair x1,x2∈A with x1<x2, it is true that f(x1 )<f(x2)
What does it mean when we say ‘f’ is monotone decreasing?
Monotone decreasing means that if whenever you move from left to right on the x-axis, the graph slopes down. Written as: for every pair x1,x2∈A with x1<x2, it is true that f(x1)>f(x2)
What does it mean when we say ‘f’ is monotone decreasing?
Monotone decreasing means that if whenever you move from left to right on the x-axis, the graph slopes down. Written as: for every pair x1,x2∈A with x1<x2, it is true that f(x1)>f(x2)
When is a function surjective?
A function is surjective if for every b in B, there exists at least on a in A such that f(a) is equal to b
What can you say about the range of a function in terms of surjectivity?
The range(f) = B
What factors influence surjectivity of a function?
The domain and codomain of a function influence surjectivity
How do I prove that a function is surjective?
You need to show that for every output there is an input, so either solve to find an input, or prove that there is an output that does not have an input
What does bijective mean?
Bijective means that a function is both injective and surjective, whereby function ‘f’ gives a complete correspondence between the elements of A and the elements of B
