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Define the first injective function, f1: A B.

Computer Science

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Answer

f1={(x,1),(y,2),(z,3)}f_1 = \{(x,1), (y,2), (z,3)\}

f2={(x,1),(y,2),(z,4)}f_2 = \{(x,1), (y,2), (z,4)\}

Step-by-step solution

2 steps

To write two injective functions from set A={x,y,z}A = \{x,y,z\} to set B={1,2,3,4}B = \{1,2,3,4\}, we need to ensure that each element in AA maps to a unique element in BB.

  1. Define the first injective function, f1:A→Bf_1: A \to B.

    We can map the elements of AA to distinct elements of BB.

    f1(x)=1f_1(x) = 1 f1(y)=2f_1(y) = 2 f1(z)=3f_1(z) = 3

    This function is injective because each element in AA maps to a different element in BB.

  2. Define the second injective function, f2:A→Bf_2: A \to B.

    To make it different from f1f_1, we can change one or more of the mappings, ensuring it remains injective.

    f2(x)=1f_2(x) = 1 f2(y)=2f_2(y) = 2 f2(z)=4f_2(z) = 4

    This function is also injective because each element in AA maps to a different element in BB.

    The two injective functions are:

    • f1:A→Bf_1: A \to B defined as f1(x)=1f_1(x) = 1, f1(y)=2f_1(y) = 2, f1(z)=3f_1(z) = 3.
    • f2:A→Bf_2: A \to B defined as f2(x)=1f_2(x) = 1, f2(y)=2f_2(y) = 2, f2(z)=4f_2(z) = 4.

    These are just two examples; many other injective functions are possible.

    The two injective functions are:

    • f1={(x,1),(y,2),(z,3)}\boxed{f_1 = \{(x,1), (y,2), (z,3)\}}
    • f2={(x,1),(y,2),(z,4)}\boxed{f_2 = \{(x,1), (y,2), (z,4)\}}
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