Equations of the null and the infinite

Sometimes an equation will resolve to eliminate all its variables, this creates two results

1.Consider

5x + 25 = 5(x-50)

5x + 25 = 5x - 50

we subtract 5x from both sides and get

25=-50 or 0=75

In this case we know the equation doesn't make sense, since 0 is NOT equal to 75. When this happens we say the equation resolves to the empty set and that there is no solution.

2. Consider

5x + 25 = 5(x+5)

5x + 25 = 5x+ 25

we subtract 5x from both sides again

25=25 or 1=1

This solution is called the identity because the left side is exactly equal to the right side. You can substitute any number for x and arrive at the same answer, thus there are an infinite number of solutions. We can say that this equation resolves to the set of all reals, often notated as R.

Hypothesis Testing: The Null and Alternative

What is a hypothesis? Any question that has a cause and effect. Statistics is often used to try answer such questions:

Does IQ relate to school grades?
Does smoking cause cancer?
Do caps protect you from aliens?

All these questions are hypothesis.

To statistically test hypothesis, they must be broken into two parts:
The null hypothesis
The alternative hypothesis

The null hypothesis always asserts the theory is false, while the alternative assumes it is true.

Null: IQ doesn't affect school grades
Alternative: IQ affects school grades

Null: Smoking doesn't cause cancer
Alternative: Smoking causes cancer

Null: Caps do not protect you from aliens
Alternative: Caps protect you from aliens

Null: Not guilty
Alternative: Guilty

Tomorrow we will look at type I and type II errors

Prove for any set A, null is a subset of A.

Prove for any set A, null is a subset of A.

Let A be any set, and x any object.

Consider the conditional sentence: x is a subset of null implies x is a subset of A. Since the antecedent (x is a subset of null) is false, then the sentence itself is true, and therefore null is a subset of A.