Symbolize the following propositional predicates 1. "All rational numbers are real numbers" 2. "Some rational numbers are not integers"

Symbolize the following propositional predicates 1. "All rational numbers are real numbers" 2. "Some rational numbers are not integers"

1.∨x( Q(x)→R(x））
2. Let R (x): X be a real number and Q (x): X be an integer
Σ (existential sign) x (R (x) ∧ Q (x))
The symbol is a little hard to type out!

Help me solve a logical predicate proof problem in Discrete Mathematics
(the individual domain is a collection of people)
If a person is afraid of difficulties, he can't succeed. Everyone succeeds or fails
Therefore, there are people who are not afraid of difficulties
How to symbolize?
Is success and failure here a right and a wrong or two different attributes?
I'll take a look at it?
Oh, I see. What do you mean by "t" and "P" in LCA? My textbooks are compiled by school teachers. The naming standard is a little different

Domain is the whole of human beings, and the predicates are defined as follows: P (x): X is afraid of difficulties; Q (x): X can succeed; R (x): X fails
The premise is symbolized as: if a person is afraid of difficulties, he can't succeed: (AX) (P (x) → not Q (x))
Everyone either succeeds or fails: (AX) (Q (x) ∨ R (x))
There are individuals who have not failed: (Ex) (not r (x))
Conclusion: there are people who are not afraid of difficulties: (Ex) (non-p (x))
(1) (Ex) (not r (x)) P
(2) Non R (a) t es (1)
(3)(Ax)(Q(x)∨R(x)) P
(4)Q(a)∨R(a) T US(1)
(5)Q(a) T(2)(4)
(6) (AX) (P (x) → non Q (x)) P
(7) P (a) → non-Q (a) t us (6)
(8) Non P (a) t (5) (7)
(9) (Ex) (non-p (x)) t eg (8)
(AX) full quantifier, (Ex) existential quantifier, P rule, t rule, ES existential designation, us full designation, eg existential generalization
To answer your supplementary, success or failure is semantically opposite, that is, failure without success, but now it is a formal proof. We can't understand it semantically without considering semantics. We can only understand it from logical composition or form. Otherwise, the premise "everyone or success or failure" is redundant, because "non-p or P" is always true and does not need to be a premise
There are two rules commonly used in formal proof, P rule and t rule. The proof process is composed of a series of formulas. Each formula occupies a single line, and the line number is added in the front of each line in order. The last line is the formula representing the conclusion, and the formulas of other lines are directly taken from the public formula in the premise (P rule), Mark the rule at the end of the line. If it is a t rule, mark which lines contain the rule and write down the line number
In the process of deduction, P rule can be directly introduced into the formula in the premise at any time
In the deduction process of t rule, the formula contained in the previous line or lines can be introduced at any time

What are the quantifiers

Quantifiers are usually used to express quantitative units of people, things or actions
Quantifiers can be divided into nominal quantifiers and momentum quantifiers
Noun quantifiers can be divided into the following categories
Proper noun quantifier
It refers to the quantifiers which have selective relations with some nouns. That is to say, some nouns can only use one or several special quantifiers. Such quantifiers are special noun quantifiers. For example, a dictionary, a horse, a fish
Temporary noun quantifier
For example: A. bring two plates of dumplings, bring a bottle of soy sauce. B. sit in a room, put a bed of things. This form generally means the amount of something in a place. There are two differences between group A and group B: 1, Generally, the numeral of group B can only be "one". 2. Group B emphasizes more quantity, but group A doesn't
Quantifier of measure name
It is mainly a unit of measurement. For example: kilogram, ruler, mu, degree, etc
Universal noun quantifier
It mainly refers to the quantifiers that are applicable to most nouns, including species, category, some and point. The Quantifier "Ge" has the tendency of generalization, and more and more nouns can be combined with it. However, there are still many nouns that can only be replaced by its special quantifier instead of "Ge", But "paper" can't say "a piece of paper". In "go, see, do and cry", the "times, times and fields" in "go, see, do and cry" are the quantitative units of action, which are called momentum words
There are two kinds of momentum words in this paragraph
Special momentum words
They include: CI, Hui, Bian, ban, Xia (ER), dun, fan. The meanings of these momentum words are different, and the ability of verb combination is also different. For example, "Ci" and "Xia (ER)" have strong ability of verb combination (that is, most verbs can be combined with them), while "ban" can only be combined with some verbs such as "go", "walk" and "run"
Instrumental momentum words
"Foot" is the tool of "kick", "slap" is the tool of "hit" and "eye" is the tool of "stare". These words are used to express the amount of action temporarily. Apart from such language environment, they are just common nouns. They can also be called temporary momentum words
Compound quantifier
The noun quantifiers "Jia", "Ren" and momentum words are used together as a special unit of measurement, that is, compound quantifiers. Although the common compound quantifiers are "noun quantifiers + momentum words" in the form, it should be noted that noun quantifiers are generally open, that is, many noun quantifiers can be used to form compound quantifiers, Some compound quantifiers are as follows: shift, person time, piece, volume, case, ship, Department, sortie, batch, household, vehicle and stage

What are the quantifiers of "paper"?
Like a piece of paper, what kind of paper Give me three more examples,

A piece of paper,
A Book of paper
A pile of paper
A stack of paper
A bundle of paper
A box of paper
A roll of paper
One page