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MATH2069: Discrete Mathematics

Creation date：2024-04-30 14:48:41

Belonging category：数学mathematics

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Discrete Mathematics

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MATH2069: Discrete Mathematics and Graph Theory Semester 1, 2022
1. Given a set X and a subset A ⊆ X , define a function fA : X → {0, 1} by setting
fA(x) = 1 if x ∈ A and fA(x) = 0 if x /∈ A. If B is also a subset of X , define
(fAfB)(x) = fA(x)fB(x).
(a) Work out fAfB when X = {1, 2, 3, 4, 5}, A = {2, 4, 5} and B = {1, 2, 5}.
Solution: fAfB takes that value 1 on 2 and 5 and is 0 elsewhere.
(b) What subset, if any, does fAfB correspond to?
Solution: fAfB corresponds to A ∩ B.
(c) Define f ′A by f
′
A(x) = 1 − fA(x). What subset, if any, does f
′
A correspond
to?
Solution: f ′A corresponds to X \ A.
(d) Form combinations of fA and fB which represent
(i) the union of A and B;
Solution: We have fA∪B(x) = fA(x) + fB(x)− fA(x)fB(x).
(ii) the symmetric difference A+B = (A \B) ∪ (B \ A) of A and B.
Solution: Define (fA + fB)(x) = fA(x) + fB(x) − 2fA(x)fB(x).
Alternatively, define (fA + fB)(x) = fA(x) + fB(x), where addition is
taken modulo 2.
2. For the following sets X , Y and function X → Y , determine whether the function
is injective or surjective
(a) X = N, Y = N, f(x) = x+ 1.
Solution: If x, y ∈ N and x + 1 = f(x) = f(y) = y + 1, then x = y.
Therefore f is injective. Recall that N = {0, 1, 2, . . .}. The function is not
surjective because x+ 1 = 0 has no solution with x ∈ N.
(b) X = Z, Y = Z, g(x) = x+ 1.
Solution: The function g : Z → Z, x '→ x + 1 is injective for the same
reason as in part (a). It is surjective because if x ∈ Z, x = (x − 1) + 1 =
g(x− 1).
(c) X = Z, Y = Z, h(x) = x2 + 5.
Solution: We show that the function h : Z → Z, x '→ x2 + 5 is neither
injective nor surjective. We have f(1) = 12 + 5 = 6 = (−1)2 + 5 = f(−1).
Furthermore, f(x) = x2+5 = 4 is insoluble because the square of an integer
is never negative.
(d) X = Z, Y = Z, p(x) = x3 + 1.
Solution: We show that the function p : Z→ Z, x '→ x3 + 1 is injective
but not surjective. First we prove that p is injective by establishing the
Copyright c© 2022 The University of Sydney 1

stronger result that p is increasing, that is, if x, y ∈ Z and x < y, then
p(x) < p(y). This follows from the fact the function of real variable f(x) =
x3 + 1 is increasing. Finally we show that p is not surjective. If n ! 2,
p(n) ! p(2) = 9. On the other hand if n " 1, p(n) " p(1) = 2. Therefore
p(n) )= 3 for any integer n.
(e) X is a nonempty set, Y = P(X) (the set of all subsets of X), h(x) = {x}.
Solution: The function h : X → P(X), x '→ {x} is injective because if
{x} = h(x) = h(y) = {y}, then x = y. However h is not surjective because
the empty set ∅ ∈ P(X) does not satisfy h(x) = {x} = ∅ for any x ∈ X .
*3. Given finite sets A and B, let E be a subset of A× B. For a ∈ A, let
E(a) = { b ∈ B | (a, b) ∈ E }
and for b ∈ B, let
E∨(b) = { a ∈ A | (a, b) ∈ E }.
Prove that ∑
a∈A
|E(a)| =
∑
b∈B
|E∨(b)|.
Solution: Find the cardinality |E|. For each a ∈ A there are |E(a)| pairs
(a, y) ∈ E and so |E| =
∑
a∈A
|E(a)|. Similarly, for each b ∈ B there are |E∨(b)|
pairs (x, b) ∈ E and therefore |E| =
∑
b∈B
|E∨(b)|. Equating these two expressions
for |E| gives the result.

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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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