Reflection of CSC165 Lec. 2 Sep. 16th.

Reflection of CSC165 Lec. 2 Sep. 16th. 

2.1 Quantifiers

2.1.1 Definition

The concept of quantifier has been mentioned in the first lecture. It is really important to remember all the definitions so I want to repeat it here as I do not think the professor did enough review on this.

"Quantifier is the expression that indicates the scope of a term to which it is attached. Moreover, quantifiers connect properties of elements to properties of sets." 

Examples of quantifiers could be all ans some.

2.1.2 Verify and Falsify

So far we have learnt two quantifiers which are universal quantification(∀) and existential quantification(∃). Verify and Falsify these quantification follow these rules as Figure 2.1 shows.

Figure 2.1

Venn diagram could visualize the problem with X and O presenting the parts is empty or occupied.

2.1.3 Quantifiers as claims about sets 

Several symbols could be used to present the relation between sets.

A ⊆ B means set A is a subset of set B. Keep in mind the symbol   ⊂ would also be used sometimes for the same meaning. Its okay to interchange these two.

A ⊄ B means set A is not a subset of set B.

A∩B means the union of set A and set B.

A∪B means the intersection between set A and set B.

A with a line on it (A-bar) means the part except A. A-bar is also refers to the average value of A sometimes, but it is not the case in terms of programming.

Ø means the empty set, where no elements are inside.

All symbols mentioned above could just be used to present the relation between two sets, so it is not correct to say a ( an element)  ⊆ A (a set) as a is just an element while ⊆ could just be used between sets. Instead, a ∈A is the correction expression.

2.1.4 Evaluating quantified claims

Using python functions to evaluate the quantified claims is somehow confused as I did not quite get the point when in the lecture, so it's important to review it again

From the statement "all employees" and "earn over $42.000", set E and O could be referred. However, it is hard to get the logic and relation between S1 and S2 from the sentence "X in S2 for X in S1".  From the answer given by professor (q3(E,O)) I could imply S2 refers to the smaller set while S1 stands for the larger set, or the statement after "for" is the larger one.

2.2 Sentences, Statements and Implications

2.2.1 Sentences V.S. Statements  

Figure 2.2 shows the brief relationship between sentences and statements, or, by using the expression studied in 2.1.4, we can say set Statement ⊆ Sentence.

Figure 2.2

More specifically,' A statement is a sentence that is NOT "open". The object in a statement is quantified, thus a statement can be evaluated true or false while a sentence can't be evaluated.'

The term quantified is the key term here. It is also suggest we can distinguish a statement by finding if it could be evaluated, but all these explanations are somehow confused as the following examples shows:

"

Exercise: Is it a statement?

➔ Someone ate my sandwich.

◆ YES

➔ The sandwich tastes delicious.

◆ NO"

I honestly think both these sentences could be evaluated by saying true or false, so the apparent different between statement and sentence is still how open the sentence is, or in other words, the quantifier. In first example, the keyword Someone leaves the sentence open as this "someone" could be anyone while in the second example, the word "the" makes the sentence stable and fixed.

With these confusions, I did more researches about sentence and statement in order to differentiate them more clearly. After asking others I know that Statement should be  a meaningful declarative sentence that is either true or false. "The sandwich tastes delicious " is not a statement because it’s a subjective personal opinion. Thus, I would like to say compare to quantified or open, subjective and objective should be the real key to distinguish statement and sentence. A statement should be a subjective fact while everyone would lead to the same result. A sentence is just a declarative description.

2.2.2 Predicate

Predicate is a Boolean-valued function P: X→ {true, false}. The concept of Boolean is also mentioned in CSC108. The result of Boolean would just be true and false so its close the property of statement.

The expression could be x (an element) ∈L(a set)

Or L(x), which means X has the property of L

We use the symbol "¬" to present the meaning neagtion. ¬L(x) means the element X does not have the property L.

Predicates without specifying X is an open sentence and it could be converted to a statement by being universally quantified or existentially quantified.

Sometimes it is hard to distinguish between a set and a predicate. Predicate is more like a property (easier to say if it is true or false.)

2.2.3 Implications

Implication works like causality. It is  presented with a antecedent and a consequent. It could also be written as P ⇒ Q which means P implies Q. The verify and falsify of an implication works the same as a universal quantifiction.

In common sense, causality should works with direct relation P causes Q, but this is not always the case in the mathematical expression. For example, we would normally say If it rains, then the ground would being wet. Here the wet ground is the direct result of the rain. However, what if we say "If it rains yesterday, then the sun rose today."  It seems ridiculous as there is no relationship between the two events, but we cannot falsify it as the sun will always rise. This implication is true mathematically.

There are two more examples to show, converse and contrapositive. Converse is the exchange between antecedent and consequent. The converse of implication may not always be true.  Contrapositive exchange the negation of antecedent and consequent. The contrapositive of a implication is exactly the same as the implication. 

2.2.4 Equivalence, sufficient and necessary

Compare to implication P ⇒ Q which is single side, Equivalence means P if and only if Q or P <=> Q.  

Necessary means P => Q, or "For P to be true, Q must to be true" while sufficient means Q => P, or the conditions P are enough to lead to result Q (Q could also be proved by other conditions).