# Are f and g Injective and Surjective if g\circf is Injective or Surjective?

• Ka Yan
In summary, the conversation discusses the effect of injectivity and surjectivity on the composition of two functions, f and g. The first judgement states that if both f and g are injective, then their composition, g\circf, is also injective. However, it is uncertain if the converse is true. Similarly, the second judgement states that if both f and g are surjective, then g\circf is also surjective, but it is unclear if the converse holds. The speaker then asks for reasons behind these judgements and suggests trying an example with a small number of elements in A, B, and C to explore the possibility of f and g not being injective but g\circf being injective.
Ka Yan
Could anybody help me check whether my judgements ture or false? (MJ = My Judgement)

Suppose f maps A into B, and g maps B into C

1. If f and g are injective, then g$$\circ$$f is injective;
(MJ)but that when g$$\circ$$f is injective, the injectivity of f and g are unsure.

2. If f and g are surjective, then g$$\circ$$f is surjective;
(MJ)and that when g$$\circ$$f is surjective, f and g are both surjective.

Thx!

Last edited:
What reasons do you have for those judgements? Can you think up a simple example where f and g are not injective but g$\circ$f is? Try the case where A, B, and C have only a few members.

1. True. If f and g are injective, then for any elements a and b in A, if f(a) = f(b), then a = b. Similarly, for any elements b and c in B, if g(b) = g(c), then b = c. This means that if g\circf(a) = g\circf(b), then a = b, making g\circf injective.

2. True. If f and g are surjective, then for any element c in C, there exists an element a in A such that f(a) = b, and for any element b in B, there exists an element c in C such that g(b) = c. This means that for any element c in C, there exists an element a in A such that g\circf(a) = c, making g\circf surjective.

## What is composite mapping?

Composite mapping is a technique used in cartography and GIS (Geographic Information Systems) to create a single map by combining multiple layers of data from different sources. This allows for a more comprehensive view of a geographic area, as well as the ability to analyze relationships between different sets of data.

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