Base cases, hypotheses and the next step.
Example 1
To prove P(n) for every integer n≥11, which pair of steps is sufficient?
Explore the structure of each case. Examples do not prove the general statement; your argument must work for an arbitrary integer.Show worked answer
Answer: Base: P(11). Step: P(k)⇒P(k+1),k≥11.
- The base case P(11) starts the chain.
- The forward implication advances it from any integer k≥11 to k+1. Both are required.
Example 3
To prove P(n) for every integer n≥9, which pair of steps is sufficient?
Explore the structure of each case. Examples do not prove the general statement; your argument must work for an arbitrary integer.Show worked answer
Answer: Base: P(9). Step: P(k)⇒P(k+1),k≥9.
- The base case P(9) starts the chain.
- The forward implication advances it from any integer k≥9 to k+1. Both are required.