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Kirsty (Parent) September 16 2016
Mathematical induction is a way of proving statements in maths. The principle is quite similar to dominoes (not pizza, the game) - if you push the first one, the second one will be pushed over, pushing the third one and so on.
There are 3 stages to induction. The first stage is to prove that the base case, n = 1 is true (essentially the first domino). The second stage involves you assuming the case n = k is true. The final stage involves you using the assumption above to show that the case n = k + 1 is also true (essentially you're showing that if you push the kth domino, the (k + 1)th domino will also be pushed over).
Here is an example of proof by induction being applied:
For any positive integer n, 1 + 2 + 3 + ... + n = n(n + 1)/2.
Let P(n) be the assumption that 1 + 2 + 3 + ... + n = n(n + 1)/2. Consider the base case, n = 1:
Left-hand side = 1. Right-hand side = 1(1 + 1)/2 = 1
As LHS = RHS, P(1) is true.
Assume that P(k) is true i.e. 1 + 2 + 3 + ... + k = k(k + 1)/2. Consider P(k + 1) [remember we have to use P(k) somewhere here]
1 + 2 + 3 + ... + k + (k + 1) = k(k + 1)/2 + (k + 1) (using P(k))
Taking a factor of (k + 1) from the RHS:
k(k + 1)/2 + (k + 1) = (k + 1)[k/2 + 1] = (k + 1)(k + 2)/2 = (k + 1)((k + 1) + 1)/2 i.e. P(k + 1) is true.
Therefore, as P(1) is true and P(k) true implies P(k + 1) is true, by the principle of mathematical induction, 1 + 2 + 3 + ... + n = n(n + 1)/2see more
Competitiveness is what is says on the tin really - if there are many firms in an industry each with a similar degree of market power, the market is competitive.
Contestability is something a little more subtle. It is the perceived threat of competition. For example, an industry may be an oligipoly, where there are only a few firms each with a large degree of market power. However, if there are low barriers to entry (i.e. low sunk costs, low costs of red tape) - basically if it's easy for new firms to enter the industry - incumbent firms will act as if there is actual competition. Contestability is all about the perceived threat of new firms entering the market and taking away some of the market power of incumbent firms.see more