Formulate mathematical relationships for a scenario involving taxis and Costa buses, considering cost and quantity constraints.

Let $x$ be the number of taxis and $y$ be the number of Costa buses.

The mathematical relationships representing the situation are:

1. Number of taxis vs. Costa buses: The number of taxis is greater than the number of Costa buses.

   $$x > y$$

2. Maximum number of taxis: The number of taxis will be less than five.

   $$x < 5$$

3. Total transport cost: The cost of hiring a taxi is Ugx 60,000, and a Costa bus is Ugx 100,000. The maximum available budget is Ugx 600,000.

   $$60,000x + 100,000y \le 600,000$$

   Dividing by 10,000:

   $$6x + 10y \le 60$$

   Dividing by 2:

   $$3x + 5y \le 30$$

4. Non-negative integer constraints: The number of taxis and Costa buses must be whole numbers and cannot be negative.

   $$x \in \mathbb{Z}, y \in \mathbb{Z}$$ $$x \ge 0, y \ge 0$$

   Given $x > y$ and $x < 5$, it implies $x$ must be at least 1. So, $x \in \{1, 2, 3, 4\}$ and $y \in \{0, 1, 2, 3\}$.

The mathematical relationships are:

$$\boxed{x > y}$$ $$\boxed{x < 5}$$ $$\boxed{3x + 5y \le 30}$$ $$\boxed{x, y \text{ are non-negative integers}}$$