Applied Mathematics - Mathematical Reasoning

Name: Applied Mathematics - Mathematical Reasoning
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Last updated 4/2022

English

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What you'll learn

Introduction
Statements
New Statements from Old
Special Words/Phrases
Implications
Validating Statements

Course content

2 sections • 33 lectures • 3h 19m total length

Introduction10:12
Introduce mathematical reasoning by defining statements as true or false, distinguishing simple and compound statements, and explaining open sentences and ambiguity with examples like Mumbai in Maharashtra.
Connectives - AND and OR8:15
Explore how simple statements form compound statements using conjunctions (and) and disjunctions (or). Learn symbolic forms such as p and q, and apply the concepts to analyze compound statements.
Connectives - IF and THEN4:21
Examine implication as an if-then conditional, dividing statements into a first part and a consequent, illustrated by 'it is raining' implies 'the match will be canceled'.
Connectives IF and ONLY IF and NEGATION7:19
Study binary connectors—conjunction, disjunction, implication, and double implication (if and only if)—and negation, using symbolic forms and the Corona will end if and only if we follow social distancing example.
Truth Table Conjunction5:54
Explore the conjunction truth table for two statements, P and Q; determine the output as true only when both are true, otherwise false, and learn symbolic compound statements.
Negation of Conjunction5:11
Explore the negation of conjunction in compound statements, showing that not (b and q) equals (not b) or (not q) via De Morgan's law, with a proof table.
Truth Table Disjunction3:48
Explore the truth table for disjunction, showing that B or Q is true if at least one is true and false when both are false, with B and Q examples.
Negation of Disjunction5:05
Apply De Morgan's law to negate a disjunction, turning not (b or q) into (not b) and (not q), with truth table verification and connections to negation of conjunction.
Truth Table Implication4:28
Explore the truth table for implication, identifying the antecedent. Note the statement is false only when the antecedent is true and the consequent is false; otherwise it is true.
Negation of Implication8:24
This lecture explains the negation of implication in logic, showing that p implies q equals not p or q, and teaches a shortcut using conjunction and negation of second statement.
Truth Table Double Implication4:24
Explore truth tables for double implication (if and only if) and learn how matching outputs yield true while differing outputs yield false, with a preview of negation.
Truth Table Negation3:22
Explore the truth table for negation in logic, showing how negation flips true and false, and apply these basics to study logical equivalence, tautology, contradiction, and contingency.
Negation of Negation4:46
Explore the negation of negation in logical statements, showing double negation cancels to yield B and a proof table with concrete examples.
Conditional Law6:49
Explore the complement law by pairing a statement with its negation, showing how B and not B yields contradiction and B or not B yields tautology, illustrated by truth tables.
Converse, Inverse and Contrapositive6:36
Explore the converse, inverse, and contrapositive of any conditional statement, learn how to form negations, and apply order changes and negation rules.
Bi Conditional Law6:48
Apply the bi conditional law to show that p double implies q is equivalent to p implies q and q implies p, using negation and disjunction forms.
Algebra of Statements Indements Law5:22
Explore the idempotent law in algebra of statements, showing that p ∧ p and p ∨ p simplify to p, using conjunction, disjunction, truth tables, and tautology concepts.
Algebra f Statements - Associative Law8:01
Explore the associative law for algebraic statements, showing that conjunction and disjunction yield the same results regardless of grouping, illustrated with truth tables and three-statement examples.
Algebra of statements - Commutative Law4:29
Explore the commutative law for statements, showing that the order of two statements in conjunction or disjunction can be swapped, with truth table justification.
Algebra of Statements - Distributive Law8:48
Apply the distributive law in the algebra of statements by distributing conjunction over disjunction, using brackets, and recognizing when to use distributive versus associative laws.
De Morgan's Law4:49
Demonstrates De Morgan's law by applying negation to each part of a conjunction or disjunction, flipping the operator, and validating with a truth table.
Identity Law10:01
Explore the identity law in logic by showing how any statement can be combined with tautology or contradiction using conjunction or disjunction, and demonstrate the behavior with truth values.
Complement Law6:48
Demonstrate the complement law by pairing a statement with its negation, producing contradiction with conjunction and tautology with disjunction, supported by truth-table reasoning.
Absorption Law5:26
Explore the absorption law in logical reasoning, showing that combining B and Q with conjunction and disjunction reduces to B, through substitution and truth-value reasoning.
Involution Law2:44
Explore the involution law in the algebra of statements, using a truth table to show that negation of negation returns the original statement.
Duality6:28
Explore duality in mathematical reasoning by transforming statements: replace conjunction with disjunction and disjunction with conjunction, and swap tautology with contradiction and contradiction with tautology.

Level - 1 Q -17:11
This lecture explains how to negate logical statements, including negation of implications, De Morgan's law, involution, and laws like associativity, complement, identity, tautology and contradiction.
Level - 1 Q -23:20
Learn to convert if-then statements into logical equivalents using not P or Q, with examples like a man is a judge implies he is honest, and root two is irrational.
Level - 1 Q -34:59
Prove the equivalence of P↔Q by applying conditional, distributive, complement, identity, and commutative laws to derive the related negations and connections between P and Q.
Level - 1 Q -44:24
Apply distributive, complement, and identity laws to combine and simplify statements, identify the common part Q, and prove B or Q without using tables.
Level - 1 Q -53:21
Learn how to construct truth tables for logical statements, including conjunctions and biconditionals, by organizing P, Q, and R values, and analyzing pattern-based outcomes.
Level - 1 Q -62:45
Construct a truth table for a two-statement pattern, derive negations and implications, and evaluate the final conjunction in a single horizontal row.
Level - 2 Solved Questions - 914:49
Solve level 2 and 3 logic questions by applying logical equivalences, negation of implies, contrapositive, De Morgan, and truth-table analysis to determine valid conclusions.

Requirements

Basic knowledge of mathematics of 9th and 10th std Mathematics

Description

Mathematical Reasoning

Mathematically acceptable statements
Connecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics
Validating the statements involving the connecting words difference between contradiction, converse and contrapositive

SUMMARY

1. A mathematically acceptable statement is a sentence which is either true or false.

2. Explained the terms:

– Negation of a statement p: If p denote a statement, then the negation of p is denoted by ∼p.

– Compound statements and their related component statements: A statement is a compound statement if it is made up of two or more smaller statements. The smaller statements are called component statements of the compound statement.

– The role of “And”, “Or”, “There exists” and “For every” in compound statements.

– The meaning of implications “If ”, “only if ”, “ if and only if ”. A sentence with if p, then q can be written in the following ways.

– p implies q (denoted by p ⇒ q)

– p is a sufficient condition for q

– q is a necessary condition for p

– p only if q – ∼q implies ∼p

– The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p . The converse of a statement p ⇒ q is the statement q ⇒ p. p ⇒ q together with its converse, gives p if and only if q.

3. The following methods are used to check the validity of statements: (i) direct method (ii) contrapositive method (iii) method of contradiction (iv) using a counter example.

Who this course is for:

Complete Mathematics for Engineering Entrance Exam Preparation. ( IIT-JEE Main | Advanced | BITSAT | SAT | etc.)
Those preparing for board and competitive exams State Board, CBSE, ICSE , IGCSE, MHT-CET & NEET
Courses are suitable for 160 countries from Europe, America, Middle East, Asia, Africa and APAC. Notably England, Germany, France, Sweden, Ireland, Scotland, USA, Canada, UAE, Saudi, Qatar, Kuwait, Malaysia, Indonesia, Myanmar, Newzealand, Australia, South Africa, South Korea, Nigeria, Nepal, Sri Lanka, etc

Applied Mathematics - Mathematical Reasoning

What you'll learn

Explore related topics

Course content

Mathematical Reasoning26 lectures • 2hr 39min

Numericals (Level 1 | Level 2 | Level 3)7 lectures • 41min

Requirements

Description

Who this course is for: