Week 6

This week, we talked about proving, non-Boolean functions, disproving a claim, and how to prove different claims.

Non - Boolean functions:
The definition of non-boolean function meant the way it sounded, functions that didn't return true of false as results. Moreover, we are introduced with a new symbol, which was the "floor x" as it was shown below.

 

As  it was shown above,  floor x must be the largest integer that was equaled or less than x, which was same as saying that floor x represents the smallest integer that was less than or equal (<=) to x.

For example, in the lecture during the week, we did various examples of simple proofs that involved floor x in the given definition and the claim. 

Given definition: 

Prove:

We proved the claim step by step by using the given definition (there are various ways of proving a claim):

 

Most of the time, we were asked to prove a claim true with a given definition, but there were occasions when we were asked to prove a claim false. This meant that we would have to "negate" the claim then prove the negation was true, and by doing this, we disproved the original claim.

Next, when proving claims that contained:

1. Conjunction (and) 

2. Disjunction (Or)

Then we would have to prove the claim by breaking the claim into different cases. 

In the following example, we were asked to prove the claim where all n that belong to natural numbers such that n^2 plus n is even.