[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