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Maths › 00. General Mathematics ›

07.5 Probabilistic and Combinatorial Proofs

7.5 Probabilistic and Combinatorial Proofs Probabilistic and combinatorial proofs establish a result by counting, arranging, or randomly selecting objects. They are especially useful for finite structures, existence statements, and identities involving numbers of configurations. A combinatorial proof shows that two expressions are equal by showing that they count the same thing. For example, the identity $$ \binom{n}{k} = \binom{n}{n-k} $$ has a direct combinatorial explanation. Choosing $k$ elements from an...

Maths › 00. General Mathematics ›

07.4 Constructive Proofs

7.4 Constructive Proofs A constructive proof proves existence by producing an object. It does not merely show that failure is impossible. It gives a witness, an algorithm, or a method that builds the required object. The basic form is: $$ \exists x \in X,\ P(x). $$ A constructive proof supplies a specific $x$ and verifies that $P(x)$ holds. For example, to prove that there exists an even prime number, we...

Maths › 00. General Mathematics ›

07.1 Direct Proof

7.1 Direct Proof A direct proof begins with the assumptions of a statement and derives the conclusion by valid steps. It follows the natural direction of implication. A typical direct proof has the form: $$ P \Rightarrow Q. $$ To prove it directly, assume $P$ and show $Q$. This method is often the clearest proof technique because it does not change the goal. The proof follows the same direction as...

Maths › 00. General Mathematics ›

07.2 Proof by Contradiction

7.2 Proof by Contradiction Proof by contradiction proves a statement by showing that its negation is impossible. Instead of proving $P$ directly, we assume $\neg P$ and derive a contradiction. The logical form is: $$ \neg P \Rightarrow \bot. $$ Here $\bot$ denotes contradiction. Once contradiction is reached, classical logic allows us to conclude $P$. This method is useful when the desired conclusion is difficult to construct directly, but the...

Maths › 00. General Mathematics ›

07.3 Induction and Recursion

7.3 Induction and Recursion Induction is the standard proof technique for objects built in stages. Recursion is the matching construction technique. Together they describe how to define, compute, and prove statements about natural numbers, lists, trees, syntax, algorithms, and other generated structures. The simplest form is mathematical induction on natural numbers. To prove a statement $P(n)$ for every natural number $n$, prove two things. First, prove the base case: $$...

Maths › 00. General Mathematics ›

07. Proof Techniques

Chapter 7. Proof Techniques Proof is the method by which mathematics separates assertion from knowledge. A statement becomes a theorem only when it follows from accepted assumptions by valid reasoning. This chapter introduces the main proof techniques used across mathematics. A proof technique is not a trick. It is a disciplined pattern of argument. Each technique is suited to a different kind of statement and a different kind of structure....

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07.5 Probabilistic and Combinatorial Proofs

07.4 Constructive Proofs

07.1 Direct Proof

07.2 Proof by Contradiction

07.3 Induction and Recursion

07. Proof Techniques