[SOLVED] COMP9020-Set 3 Functions and Relations

Description

1. Functions)

Let                                         . Consider the functions  given by

Compute the following function values:

2. (Properties of functions)

Which of the three functions ,  and  in Exercise 1 is onto? Which are 1-1?

3. (Matrix functions)

Prove each of the following statements.

1. (A^T)^T = A for any matrix A.
2. If two matrices A and B are of the same size, then (A + B)^T = A^T + B^T.
3. A(B + C) = AB + AC for any matrix A of size m × n and matrices B, C of size n × p.

4. (Boolean functions)
   1. Give all elements of BOOL(2), that is, all functions over two Boolean variables.
   2. Show that there are elements in BOOL(n) for
5. (Properties of binary relations)

Reflexivity

Antireflexivity

Symmetry

Antisymmetry

Transitivity

1. For each of the following statements, give a valid proof if it is true for all relations and over arbitrary sets . If the statement is not always true,

provide a counterexample.

If        and          are symmetric, then                  is symmetric.

If        and          are antisymmetric, then       is antisymmetric.

6. Challenge Exercise

Consider a set         and the binary relation                                 defined by .

Prove that        is transitive if and only if