Linear Algebra II

August 30, 2014

Logical Warm-up

Filed under: 2014 Fall — Y.K. Lau @ 4:49 PM

Someone said, “If mathematics is regarded as a language, then logic is its grammar.” This is true and that’s why we have the course MATH1001/2012 — Fundamental concepts of mathematics. Here we start with an introductory remark on “logic”, serving as a warm-up and a preliminary.

1. Statements

A statement is a declarative sentence, conveying a definite meaning that may be either true or false but not both simultaneously.

Example 1 Which of the following is a statement?

1. Every HKU undergraduate student has a university number.
2. There is a man who is over six feet tall.
3. Linear algebra smells good.
4. If there is life on Mars, then the postman delivers letters.
5. This sentence is false.

Ans.

Clearly, (1) and (2) are statements. Definitely (2) is a true statement. For (1), it is either true or false, although frankly I don’t know the answer because I haven’t checked through every student of HKU.

(3) is not a statement. (Does linear algebra have odor?)

(4) is a statement and it is a true statement, because the job of postman is to deliver letters. So no matter whether or not Mars has life, the conclusion is valid.

(5) is a bit more tricky — it is not a statement. The reason is that you cannot assign a true/false value to it. Why? Think about what it means if (5) is true — it says “this sentense, i.e. (5), is false”. So (5) is true and false at the same time! If (5) is false, that means “This sentence is false” is not valid. In other words, this sentence (5) is true. Again (5) is false and true simultaneously. That’ why we cannot give it a true/false value.

Important Remark

1. TRUE means absolutely and completely true. There is no in-between stage. So there is no such thing as saying a statement is somewhat true’ or almost true’.
2. For some statements, you may pay careful attention to what member of a class the statement applies. For statement (1) of Example 1, a single exception will renders false. But for (2), I am shorter than six-feet (and I am a man). This fact of my height cannot conclude statement (2) is false. We call “for all/for every/for any” and “there exists/for somequantifiers, which will arise often in our course.
3. The truth value of a statement may depend upon the way it is interpreted. For example, “you are intelligent”. I would say that it is a true statement (because you take MATH1111), but you may say it is a false statement. (You are humble!) But in any case, once intelligence is clearly defined, then the statement must be declared as either completely true or false.

2. Negation

Negation turns a statement into another statement which will be opposite to the original one in terms of truth value. For example, “X is rich”. Its negation is “X is poor”. It sounds easy. Well, finish the example below.

Example 2 Write down the negation of each of the following statements.

1. Some men are rich.
2. Every man is happy.
3. There exists a man who is rich.
4. Some unhappy men are rich.
5. There is a happy man that all of his friends are unhappy.

If you have finished, see answers below.

Ans.

1. All men are poor. (I suppose the negation of rich (i.e. not rich) is the same as poor.)
2. Some men are unhappy. (Think carefully if your answer is “Every man is unhappy” or “No man is happy”, which are not correct.)
3. All men are poor. (Because “There exists a man who is rich” means the same as “Some men are rich”.)
4. All unhappy men are poor. (The answer is not “All happy men are poor”, because for the given statement “Some unhappy men are rich”, we are considering the group of “unhappy men” and the statement says that some members in this group are rich. So the negation is “All member in this group are poor”.)
5. Every happy man has some happy friends. Alternatively, the negation can be stated as “For any happy man, there exists some friends of him who are happy.”

3. Logical Implications

Here I would not tell precisely what is a logical implication. Instead, let me introduce some notation and terminologies. We write “${p\Rightarrow q}$” when the statement ${p}$ implies (affirmatively) the statement ${q}$“. What does “${p\Rightarrow q}$” tell? There are two important points:

1. If ${p}$ is true, then ${q}$ is true.
2. If ${q}$ is false, then ${p}$ is also false.

Let us look at an example.

Example 3

1. ABC is a triangle ${\Rightarrow}$ ${\angle A + \angle B+ \angle C = 180}$ degrees. (I think you know this fact. This is an example of point 1.)
2. ${\sin \frac{\pi}4 >\frac45}$ ${\Rightarrow}$ ${-1>0}$. (Refer to point 2. Hence we know ${\sin \frac{\pi}4 >\frac45}$ is false.)

You may wonder how the deduction in (ii) comes up. Below is the details:

${\sin \frac{\pi}4 >\frac45\Rightarrow \sin^2 \frac{\pi}4 >\frac{16}{25}\Rightarrow 1-2\sin^2\frac{\pi}4 < 1-\frac{32}{25}\Rightarrow \cos \frac{\pi}2 < -\frac{7}{25} \Rightarrow 0 <-1.}$

Remark. When the statement ${p}$ in “${p\Rightarrow q}$” is false, we cannot draw any conclusion on ${q}$, i.e. ${q}$ may be true or not. For example,

$\displaystyle 1\ge 2 \Rightarrow -1 \ge 0.$

(This is deduced by subtracting 2 on both sides of $\displaystyle 1\ge 2.$) The consequence ${-1\ge 0}$ is of course a false statement. However,

$\displaystyle 1 \ge 2 \Rightarrow 0\ge 0.$

(We have multiplied ${0}$ on both sides of ${1\ge 2}$.) Now the consequence ${0\ge 0}$ is a true statement! So we don’t know the truth value of the conclusion from a false hypothesis.

There are other ways to phrase “${p}$ implies ${q}$“:

1. ${p}$ is a sufficient condition for ${q}$.
2. ${q}$ is a necessary condition for ${p}$.
3. ${p}$ only if ${q}$.

4. Final Remark

Logic is, but not just confined to be, the grammar of MATHEMATICS. For more, you may browse the following website:

http://philosophy.hku.hk/think/logic/intro.php