International Journal of Trend Volume 5 Issue 6, September-October @ IJTSRD | Unique Paper ID – IJTSRD The Use of Logic Science Elements in the Preparation of Students with Creative Abilities K. B. Andijan State University ABSTRACT This article explains the importance of using the algebraic formulas of predicates in teaching students proof of theorems and solution of inequalities. From the examples and theorems presented in the article, it is possible to use them not only in teaching the exercises of the algebraic of predicates, but also in working with students with creative organizing classes on the subject of mathematics in circles. The science of mathematics is studied and developed on the basis of the laws of mathematical logic. In this case, the science of mathematical logic is not taught as a separate science in secondary schools. Although elements of mathematical logic are partially included in the textbooks of Mathematics, its applications are not sufficiently covered. As a result, students are faced with many difficulties in an in-depth study of the theoretical foundations of mathematical science, solving equations and inequalities, especially in proving theorems. Taking this into account, in this article we will consider the applications of the algebra of predicates to solve the system of inequalities and inequalities, as well as to prove theorems In the study of the application predicates, it is important to know its equivalent formulas. Let's mention the main equivalent formulas of algebraic predicates:
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⋁⋀ in Scientific Research and Development 2021 Available Online: www.ijtsrd.com 47691 | Volume – 5 | Issue – 6 | Sep-Oct Mamadaliyev, B. K. Mamadaliyev , Andijan, Uzbekistan abilities, and also in How to cite Mamadaliyev | B. K. Mamadaliyev "The Use of Logic Science Elements in the Preparation of Students with Creative Abilities" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456 6470, Volume Issue-6, October 2021, pp.1601 URL: www.ijtsrd.com/papers/ijtsrd47691.pdf Copyright © International Scientific Research Journal. This distributed under terms of the Commons Attribution License (http://creativecommons.org/licenses/by/4.0 of algebraic ⋀ (1) 
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⋁ Since inequalities consist of predicates, the of solving inequalities comes to the question of finding the sphere of truth of the predicate. Let P(x) and S(x) be the predicates defined in an M set. We denote the areas of truth of these predicates with E and Es respectively, the denominator o predicate P(x) and the collection M use the following formulas when finding the areas of the predicate
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 ⟺ ̅   . ∨   ∪ . ∧   ∩ . (IJTSRD) e-ISSN: 2456 – 6470 2021 Page 1601 this paper: K. B. - -5 | -1604, 2021 by author (s) and Journal of Trend in and Development is an Open Access article the Creative (CC BY 4.0) )  (3)  (4)  (5)
 (6) ⋁
⋀̅ (7) ⋀̅⋁
 (8) question p f the \Epwith  .We
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⋀,  in the predicate. (9) (10) (11) IJTSRD47691 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47691 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1602 ⟹   ∪  (12) ⟺   ∩  ∩   ∩   (13) ⟺    ∪  ∩   ∪ . (14) Proof of these formulas comes from the definitions of actions on predicates and the above-mentioned equivalent formulas of algebraic predicates [4]. R-let it be a set of real numbers. 1-example. x2-4x-5<0 predicate in R set is given. Find the area of its truthfulness. Solution. We mark the given predicate with P(x), the sphere of its truthfulness with Ep. Thus,
 ≡  − 4 − 5 < 0 ≡  + 1 − 5 < 0 ≡ ≡  + 1 < 0 ∧  − 5 > 0 ∨  + 1 > 0 ∧  − 5 < 0 ≡ ≡  < −1 ∧  > 5 ∨  > −1 ∧  < 5, From this and (10), (11) we find,   −∞; −1 ∩ 5;∞ ∪ −1;∞ ∩ −∞; 5  ∅ ∪ −1; 5  −1; 5. Answer:  −1; 5. 2-example. P(x)=(x2-x-20>0) predicate in R set is given. Find the area of its truthfulness. Solution.
 ≡  −  − 20 > 0 ≡  + 4 ⋅  − 5 > 0 ≡ ≡  + 4 < 0 ∧  − 5 < 0 ∨  + 4 > 0 ∧  − 5 > 0 ≡ ≡  < −4 ∧  < 5 ∨  > −4 ∧  > 5, From this and (10), (11) we find,   & ∈ (| < −4* ∩ & ∈ (| < 5* ∪ ∪ & ∈ (| > −4* ∩ & ∈ (| > 5*  −∞; −4 ∩ −∞; 5 ∪ ∪ −4;∞ ∩ 5;∞  −∞; −4 ∪ 5;∞ . Answer:  −∞; −4 ∪ 5;∞. 3-example.
  + ,-. /,012 ≤ 04predicate in R set is given. Find the area of its truthfulness. Solution.  + ,-. /,012 ≤ 04 ≡ 2 + 6 ≤ 0 ∧ 5 − 10 > 0 ∨ ∨ 2 + 6 ≥ 0 ∧ 5 − 10 < 0 ≡  ≤ −3 ∧  > 2 ∨  ≥ −3 ∧  < 2 . From this and (10), (11) we find,  & ∈ (| ≤ −3* ∩ & ∈ (| > 2* ∪∪ & ∈ (| ≥ −3* ∩ & ∈ (| < 2*  −∞; −38 ∩ 2;∞ ∪ 9−3;∞ ∩ ∩ −∞; 2  ∅ ∪ 9−3; 2  9−3; 2. Answer:  9−3; 2. 4-example.P(x)=(|x-2|<3) predicate in R set is given. Find the area of its truthfulness. Solution.
  | − 2| < 3 ≡  − 2 < 3 ∧  − 2 > −3 ≡ ≡  < 5 ∧  > −1. From this and (11),   & ∈ (| < 5 ∧  > −1*  & ∈ (| < 5* ∩ & ∈ (| > −1*   −∞; 5 ∩ −1;∞  −1; 5. Answer:  1; 5. 5-example.
  |2 + 6| ≥ 4predicate in R set is given. Find the area of its truthfulness. Solution.
  |2 + 6| ≥ 4 ≡ 2 + 6 ≥ 4 ∨ 2 + 6 ≤ −4 ≡ ≡ 2 ≥ −2 ∨ 2 ≤ −10 ≡  ≥ −1 ∨  ≤ −5. From this and (11),   & ∈ (| ≥ −1 ∨  ≤ −5*  & ∈ (| ≥ −1* ∪ & ∈ (| ≤ −5*   9−1;∞ ∪ −∞; −58  −∞; −58 ∪ 9−1;∞, Answer:  −∞; −58 ∪ 9−1;∞. 6-example. R is given the predicates    −  ≤ 0and    ≤ √, which are determined in the set.  ?,  ?, ∧ ?, ∨ ?, ⟹ ?, ⟹ ?, ⟺ ? Find. Solution. (9) - (14) we use formulas.   & ∈ (| −  ≤ 0*  & ∈ (| − 1 ≤ 0*  & ∈ (| ≤ 0 ∧ ∧  − 1 ≥ 0* ∨  − 1 ≤ 0 ∧  ≥ 0*  & ∈ (| ≤ 0* ∩ & ∈ (| ≥ 1* ∪ ∪ & ∈ (| ≤ 1* ∩ & ∈ (| ≥ 0*  −∞; 08 ∩ 91;∞ ∪ −∞; 18 ∩ 90;∞   ∅ ∪ 90; 18  90; 18 ;   90; 18 .   < ∈ (= ≤ √>  & ∈ (| ≥ 0 ∧  ≤ *  & ∈ (| ≥ 0* ∩ ∩ & ∈ (| − 1 ≤ 0*  90;∞ ∩ 90; 18  90; 18.   90; 18 . ∧   ∩   90; 18 ∩ 90; 18  90; 18 ∨   ∪   90; 18 ∪ 90; 18  90; 18 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47691 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1603 ⟹    ∪   −∞; 0 ∪ 1;∞ ∪ 90; 18  −∞;∞. ⟹  −∞;∞. ⟺  ⟹ ∩ ⟹  −∞;∞. The following theorems can be used in teaching students how to solve proof-of-concept issues using equivalent formulas of predicate algebra. 1. theorem.∀ ∈ ( ≤  ⟹  ≤ √. 2. theorem. ∀ ∈ ( ≤ √ ⟹  ≤ . 3. theorem. ∀ ∈ ( ≤ √ ⟺  ≤ . 4. theorem. ∀ ∈ ℳ
 ⟹  ⟹  ⊂ . 5. theorem.  ⊂  ⟹ ∀ ∈ ℳ
 ⟹ . 6. theorem. ∀ ∈ ℳ
 ⟺  ⟹   . 7. theorem.    ⟹ ∀ ∈ ℳ
 ⟺ . This is easily proved by the method of hypothesis of the opposite of theorems. We are limited to bringing proof of the 7-theorem. Proof. We use the method of hypothesis from the opposite. ∀ ∈ ℳ
 ⟺  ≡ ∃ ∈ ℳ
 ⟺  ≡ ≡ ∃ ∈ ℳ
 ∧  ∨
 ∧ ̅ ≡ ≡ ∃ ∈ ℳ
 ∨ ̅ ∧ 
 ∨  ≡ ≡ ∃ ∈ ℳ
 ∧  ∨ ̅ ∧
, ∃ ∈ ℳ
 ∧   1 ∨ ̅ ∧
  1 ≡ ∃ ∈ ℳ
  1 ∧   1 ∨ ̅  1 ∧ 
  1 ≡ ∃ ∈ ℳ
  0 ∧   1 ∨   0 ∧ 
  1 ≡ ∃ ∈ ℳ ∈   ∧  ∈  ∨  ∈   ∧  ∈  ⟹ ⟹    ⟹  ≠ . The theorem was proof. 8- theorem. ⟺   ∩  ∩   ∩   Proof. x element ⟺ balllamga get the appropriate optional element. ⟹
 ∧  ∨
 ∧ ̅  1 ⟹ ⟹ 
 ∧   1 ∨ 
 ∧ ̅  1 ⟹ ⟹ 
  1 ∧   1 ∨ 
  1 ∧ ̅  1 ⟹ ⟹  ∈  ∧  ∈  ∨  ∈ ̅ ∧  ∈ ̅ ⟹ ⟹  ∈  ∩  ∨  ∈   ∧  ∈   ⟹ ⟹  ∈  ∩  ∨  ∈   ∩   .  ∈  ∩  ∪   ∩   .  ∈  ∩  ∨  ∈   ∩   ⟹ ⟹  ∈  ∧  ∈  ∨  ∈   ∧  ∈   ⟹ ⟹ 
  1 ∧   1 ∨  ∈ ̅ ∧  ∈ ̅ ⟹ ⟹ 
  1 ∧   1 ∨ 
  1 ∧ ̅  1 ⟹ ⟹ 
 ∧   1 ∨ 
 ∧ ̅  1 ⟹ ⟹
 ∧  ∨
 ∧ ̅  1 .
 ⟺   1 ⟹  ∈ ⟺. The theorem was proof. 9-theorem.⟺    ∪  ∩   ∪ . Proof.
 ∨  ∧ ̅ ∨
  1 ⟹ ⟹ 
 ∨   1 ∧ ̅ ∨
  1 ⟹ ⟹ 
  1 ∨   1 ∧ ̅  1 ∨ 
  1 ⟹ ⟹  ∈ ̅ ∨  ∈  ∧  ∈ ̅ ∨  ∈  ⟹ ⟹  ∈   ∨  ∈  ∧  ∈   ∨  ∈  ⟹ ⟹  ∈   ∪  ∧  ∈   ∪  ⟹  ∈   ∪  ∩   ∪ .  ∈   ∪  ∩   ∪ ,  ∈   ∪  ∩   ∪  ⟹  ∈   ∪  ∧  ∈   ∪  ⟹ ⟹  ∈   ∨  ∈  ∧  ∈   ∨  ∈  ⟹ ⟹  ∈ ̅ ∨  ∈  ∧  ∈ ̅ ∨  ∈  ⟹ ⟹ 
  1 ∨   1 ∧ ̅  1 ∨ 
  1 ⟹ ⟹ 
 ∨   1 ∧ ̅ ∨
  1 ⟹ ⟹ 
 ∨  ∧ ̅ ∨
  1. 
 ⟺   1 ⟹  ∈ ⟺.The theorem was proof. Of the examples and issues discussed above, it is possible to use them in teaching students the International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47691 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1604 application of predicate algebra. The students are taught the laws of the science of mathematical logic, the rules of derivation, equivalent formulas and their applications in depth and in detail, their ability to solve problematic situations quickly and without errors develops. Used literature [1] K. B. Mamadaliyev, B. K. Mamadaliyev. Use of Elements of Predicate Algebra in Solving Proof Problems. Central asian journal of theoretical and applied sciences. Volume: 02 Issue: 08 | Aug 2021 ISSN: 2660-5317. https://cajotas.centralasianstudies.org/index.p hp/CAJOTAS/article/view/219. [2] K. B. Mamadaliyev, B. K. Mamadaliyev. Solving problems of applications of collection theory. academicia globe: Inderscience Research. issn: 2776-1010 Volume 2, Issue 5, May, 2021. https://agir.academiascience.org/index.php/ag ir/article/view/126/115