The function \(g:\mathbb{Z} \to \mathbb{Z}\) is defined by \(g(n) = \left| n \right| - 1{\text{ for }}n \in \mathbb{Z}\) . Show that g is neither surjective nor injective.

The set S is finite. If the function \(f:S \to S\) is injective, show that f is surjective.

Using the set \({\mathbb{Z}^ + }\) as both domain and codomain, give an example of an injective function that is not surjective.

non-S: for example –2 does not belong to the range of g     R1

non-I: for example \(g(1) = g( - 1) = 0\)     R1 

Note: Graphical arguments have to recognize that we are dealing with sets of integers and not all real numbers

[2 marks]

as f is injective \(n\left( {f(S)} \right) = n(S)\)     A1

     R1

Note: Accept alternative explanations.

f is surjective     AG

[2 marks]

for example, \(h(n) = n + 1\)     A1 

Note: Only award the A1 if the function works.

I: \(n + 1 = m + 1 \Rightarrow n = m\)     R1

non-S: 1 has no pre-image as \(0 \notin {\mathbb{Z}^ + }\)     R1

[3 marks]

Nearly all candidates were aware of the conditions for an injection and a surjection in part (a). However, many missed the fact that the function in question was mapping from the set of integers to the set of integers. This led some to lose marks by applying graphical tests that were relevant for functions on the real numbers but not appropriate in this case. However, many candidates were able to give two integer counter examples to prove that the function was neither injective or surjective. In part (b) candidates seemed to lack the communication skills to adequately demonstrate what they intuitively understood to be true. It was usually not stated that the number of elements in the sets of the image and pre – image was equal. Part (c) was well done by many candidates although a significant minority used functions that mapped the positive integers to non – integer values and thus not appropriate for the conditions required of the function.

Nearly all candidates were aware of the conditions for an injection and a surjection in part (a). However, many missed the fact that the function in question was mapping from the set of integers to the set of integers. This led some to lose marks by applying graphical tests that were relevant for functions on the real numbers but not appropriate in this case. However, many candidates were able to give two integer counter examples to prove that the function was neither injective or surjective. In part (b) candidates seemed to lack the communication skills to adequately demonstrate what they intuitively understood to be true. It was usually not stated that the number of elements in the sets of the image and pre – image was equal. Part (c) was well done by many candidates although a significant minority used functions that mapped the positive integers to non – integer values and thus not appropriate for the conditions required of the function.

Nearly all candidates were aware of the conditions for an injection and a surjection in part (a). However, many missed the fact that the function in question was mapping from the set of integers to the set of integers. This led some to lose marks by applying graphical tests that were relevant for functions on the real numbers but not appropriate in this case. However, many candidates were able to give two integer counter examples to prove that the function was neither injective or surjective. In part (b) candidates seemed to lack the communication skills to adequately demonstrate what they intuitively understood to be true. It was usually not stated that the number of elements in the sets of the image and pre – image was equal. Part (c) was well done by many candidates although a significant minority used functions that mapped the positive integers to non – integer values and thus not appropriate for the conditions required of the function.