CBSE Class 11 Maths Mathematical Reasoning – Get here the Notes for Class 11 Mathematical Reasoning. Candidates who are ambitious to qualify the Class 11 with good score can check this article for Notes. This is possible only when you have the best CBSE Class 11 Maths study material and a smart preparation plan. To assist you with that, we are here with notes. Hope these notes will helps you understand the important topics and remember the key points for exam point of view. Below we provided the Notes of CBSE Class 11 Maths for topic Mathematical Reasoning.
- Class: 11th
- Subject: Maths
- Topic: Mathematical Reasoning
- Resource: Notes
CBSE Notes Class 11 Maths Mathematical Reasoning
Candidates who are pursuing in CBSE Class 11 Maths are advised to revise the notes from this post. With the help of Notes, candidates can plan their Strategy for particular weaker section of the subject and study hard. So, go ahead and check the Important Notes for CBSE Class 11 Maths Mathematical Reasoning from this article.
Logic
Logic is the subject that deals with the method of reasoning. It provides us rules for determining the validity of a given argument in proving theorem.
Statement (Proposition)
A statement is an assertive sentence which is either true or false but not both a true statement is called valid statement. Otherwise it is called invalid statement. Statements are denoted by the small letters
i.e., p, q, r … etc.
Open and Compound Statement
A sentence which contains one or more variable such that when certain values are given to the variable it becomes a statement, is called an open statement.
If two or more simple statements are combined by the use of words such as ‘and’, ‘or’, ‘not’, ‘if’, ‘then’, ‘if and only if ‘, then the resulting statement is called a compound statement.
Truth Value and Truth Table
A statement can be either ‘true’ or ‘false’ which are called truth values of a .statement and it is represented by the symbols T and F, respectively.
A. truth table is a summary of truth values of the resulting statements for all possible assignment of values to the variables appearing in a compound statement.
Number of rows depends on their number of statements.
Truth Table for Two Statements (p, q)
Number of rows = 22 = 4
| p | q |
| T | T |
| T | F |
| F | T |
| F | F |
Elementary Operation of Logic
(i) Conjunction A compound sentence formed by two simple sentences p and q using connective ‘and’ is called the conjunction of p and q and it is represented by p ∧ q.
(ii) Disjunction A compound sentence formed by two simple sentences p and q using connectives ‘or’ is called the disjunction of p and q and it is represented by p ∨ q.
(iii) Negation A statement which is formed by changing the truth value of a given statement by using the word like ‘no’, ‘not’ is called negation of given statement. If p is a statement, then negation of p is denoted by ~ p.
(iv) Conditional Sentence (Implication) Two simple sentences p and q connected by the phase, if and then, is called conditional sentence of p and q arid it is denoted by p => q.
(v) Biconditional Sentence (Bi-implication) The two simple sentences connected by the phrase, ‘if and only if this is called biconditional sentences. It is denoted by the symbol ‘<=>’.
Table for Basic Logical Connections
Tautology and Contradiction
The compound statement which are true for every value of their components are called tautology.
The compound statements which are false for every value of their components are called contradiction (fallacy).
Truth Table
Quantifiers and Quantified Statements
In this statements, there are two important symbols are used.
(i) The symbol ‘∀’ stands for ‘all values of ‘.
This is known as universal quantifier.
(ii) The symbol ‘∃’ stands for ‘there exists’.
The symbol ∃ is known as existential quantifier.
Quantified Statement
An open sentence with a quantifier becomes a statement, called a quantified statement.
Negation of a Quantified Statement
(i) ~{∀ x ∈ A : p(x) is true} = {∃ x ∈ A such that (s.t.) ~ p(x) is true}
(ii) ~{∃x ∈ A : p(x) is true} = {∀ x ∈ A : ~ p(x) is true}
Laws of Algebra of Statements
(i) Idempotent Laws
(a) p ∨ p ≡ p
(b) p ∧ p ≡ p
(ii) Associative Laws
(a) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(b) (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
(iii) Commutative Laws
(a) p ∨ q ≡ q ∨ p
(b) p ∧ q ≡ q ∧ P
(iv) Distributive Laws
(a) p ∨ (q A r) ≡ (p ∨ q) ∧ (p ∨ r)
(b) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
(v) De Morgan’s Laws
(a) ~(p ∨ q) ≡ (~ p) ∧ (,_ q)
(b) ~(p ∧ q) ≡ (~ p) ∨ (~ q)
(vi) Identity Laws
(a) p ∧ F ≡ F
(b) p ∧ T ≡ p
(c) p ∨ T ≡ T
(d) p ∨ F ≡ p
(vii) Complement Laws
(a) p ∨ (~ p) ≡ T
(b) p ∧ (~ p) ≡ F
(c) ~ (~p) ≡ p
(d) ~ T ≡ F, ~ F ≡ T
Important Points to be Remembered
(i) The number of rows of table is depend on the number of statements.
(a) If p is false, then ~ p is true.
(b) If P is true, then ~ p is false.
(ii) (a) The converse of p => q is q => p.
(b) The inverse of p => q is ~ p => ~ q.
(iii) The contrapositive of p => q is ~ q => ~ p.
A statement which is neither a tautology nor a contradiction is a contingency.
Class 11 Key Points, Important Questions & Practice Papers
Hope these notes helped you in your schools exam preparation. Candidates can also check out the Key Points, Important Questions & Practice Papers for various Subjects for Class 11 in both Hindi and English language form the link below.
Class 11 NCERT Solutions
Candidates who are studying in Class 11 can also check Class 11 NCERT Solutions from here. This will help the candidates to know the solutions for all subjects covered in Class 11th. Candidates can click on the subject wise link to get the same. Class 11 Chapter-wise, detailed solutions to the questions of the NCERT textbooks are provided with the objective of helping students compare their answers with the sample answers.
Class 11 Mock Test / Practice
Mock test are the practice test or you can say the blue print of the main exam. Before appearing in the main examination, candidates must try mock test as it helps the students learn from their mistakes. With the help of Class 11 Mock Test / Practice, candidates can also get an idea about the pattern and marking scheme of that examination. For the sake of the candidates we are providing Class 11 Mock Test / Practice links below.
Class 11 Exemplar Questions
Exemplar Questions Class 11 is a very important resource for students preparing for the Examination. Here we have provided Exemplar Problems Solutions along with NCERT Exemplar Problems Class 11. Question from very important topics is covered by Exemplar Questions for Class 11.
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