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Let f : AB and g : BC denote two functions. Consider the following two statements:
S1 : If both f and g are injections then the composition function : AC is an injection.
S2 : If the function : A → C is surjection and g is an injection then the function f is a surjection.
S3 : If h(a) = g(f(a)) and h(a) is onto then g must be onto, where ∀a, aA.
Which of the above statements are valid?

asked in Mathematical Logic by Boss (8.6k points)   | 64 views
all valid?
I couldnt find any contradicting example.
What is your approach to above problem?

How did you started?
Just took 3 sets with cardinalities 3/4 and try to have simple mappings. Just you need to check if you can contadict the statement. There is no specific rule . I hope you get it.
please explain 3rd statement

There is some function h: A->C which is onto. 

Given that h(a) = g( f(a) ), try to find a function 'g' which isnt onto. If you can find, then you have contradicted the statement.

1 Answer

+1 vote

Let f : A → B and g : B → 

S1: if f and g are injection fuction then composition function  gof :A → C is an injection. ----> this statement true its well known property.

S2:If the function gof : A → C is surjection and g is an injection then the function f is a surjection.

here keypoint is g is injection fuction.

means that  g(f(x) = g(y) -----> f(x) = y  so that fuction f is surjection

S3:If h(a) = g(f(a)) and h(a) is onto then g must be onto, where ∀aa ∈ A

all of the statement true. its all property .

answered by Boss (6.3k points)  

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