Direct proof is one the easiest method to construct proofs.
To use this method, we must prove that a theorem, stated as a conditional statement p-> q is true.
To this, we assume that p is true, and we apply rules of inference, axioms, definitions and previously proved theorems to prove that q is also true. By doing this, we warrantee that truth values of p true and q false never occur.
Let’s see the solution to this example.
We can restate the conjecture we need to prove as follows:
If m and n are odd integers, then m+n is even.
First, let’s use the definition of and odd number.
If n is odd, then n = 2k +1 for some integer k.
m = 2k +1, by definition of odd number
n = 2t +1, by definition of odd number
By doing this, we assumed that p is true. In other words, that m and n are odd integers. Notice that we express m by using k, and n by using t. We shouldn’t assume that k=t. Although it is a possibility, it doesn’t cover all the cases.
m+n = 2k+1 + 2t + 1
Now we should proof that m+n is an even number.
By definition, an even integer p = 2r for some integer r.
m+n = 2k+1 + 2t + 1 = 2k + 2t + 2 = 2(k+t+1) ∎
Let me summarize the steps used to create this proof:
- Write the conjecture as a conditional statement p -> q. Remember a conjecture is a statement that we think is true based on some evidence. Once a conjecture is proved, it is a theorem.
- Assume p is true. In this case, we assumed p is true by expressing m and n as even numbers. We did this, by using the even number definition.
- We applied definitions to add m+n. The we used basic arithmetic rules to transform m+n.
- Using the definition of even number, we proved that m+n is even.
- Use a direct proof to show that the sum of two even integers is even
- Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational
- Prove or disprove that the product of two irrational numbers is irrational
- Show that if n is an integer and n^3+5 is odd, then n is even using a proof by contraposition and a proof by contradiction
- Prove that if m and n are integers and mn is even, then m is even or n is even