# Algorithm Design Manual Problem 1–1

A lighter problem from the Algorithm Design Manual (second edition).

### Problem

Show that a + b can be less than min(a, b).

### Thoughts

- a + b < min(a, b)
- If a < b : a + b < a
- If b < a : a + b < b

#### Cases

*a < b*

1.*a + b < a*

2.*b < a — a*

3.*b < 0**b < a*1.*a + b < b*2.*a < b — b*3.*a < 0*

### Solution

If `a` and `b` are less than `0` then the addition of both is less than the minimum of the pair.

#### Example

a <- (-2)

b <- (-3)

minimum <- min(a, b) // (-3)

sum <- (-2) + (-3) // (-5)

sum < minimum // Matches condition for completion