Please Help. Thank you! For each of the functions below, determine and prove whether or not it is injective, surjective, andbijective.41. ƒ : {0, 1}³ → {0, 1}ª is given by adding a copy of the first bit to the end of the binary string.In other words f(xyz) = xyzx.2. Let S = {1, 2, 3} and consider g: P(S) → {0, 1, 2, 3} given by g(A) = |A|, where recall thatfor any set A, |A| denotes its cardinality.

Answered: For each of the functions below,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please Help. Thank you!

For each of the functions below, determine and prove whether or not it is injective, surjective, and
bijective.
4
1. ƒ : {0, 1}³ → {0, 1}ª is given by adding a copy of the first bit to the end of the binary string.
In other words f(xyz) = xyzx.
2. Let S = {1, 2, 3} and consider g: P(S) → {0, 1, 2, 3} given by g(A) = |A|, where recall that
for any set A, |A| denotes its cardinality.

Expert Solution

Step 1

1. f is injective. To see this is true, let us consider a, b in ${0, 1}^{3}$ . Assume that f(a)=f(b). This means that all four coordinates of f(a) and f(b) are equal. This means that the first three coordinates are equal and are in the same order. This means that a=b, meaning f is injective.

f is not surjective. To see this, a sequence of the form $x y z y \in {0, 1}^{4}$ . But that element has got no preimage.

So, f is not bijective as f is not surjective.