[SOLVED] COMP9020 Week 3- Functions and Relations

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.

4. (A^T)^T = A for any matrix A.
5. If two matrices A and B are of the same size, then (A + B)^T = A^T + B^T.
6. A(B + C) = AB + AC for any matrix A of size m × n and matrices B, C of size n × p.
7. (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
8. (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