Mathematical Logic — Class 12 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Mathematical Logic, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Mathematical Logic in one look
statements, connectives, and the algebra that simplifies both circuits and arguments
Statements and Logical Connectives
- a statement (proposition) is a sentence, definitely true or false, never both
- statements are denoted p, q, r, ...
- questions, commands, requests, exclamations are never statements
- an open sentence like x+6=9 isn't one either, its truth changes with x
- a simple statement can't split into smaller ones
- a compound statement joins two or more simple ones with a connective
- 'and' gives conjunction p∧q
- 'or' gives disjunction p∨q
- 'if...then' gives conditional p→q
- 'if and only if' gives biconditional p↔q
- 'not' gives negation ~p
Truth Tables
- Conjunction, p∧q: true only when both are true: T∧T≡T, else F
- Disjunction, p∨q: false only when both are false: F∨F≡F, else T
- Conditional, p→q: false only when p is true, q is false: T→F≡F, else T
- Biconditional, p↔q: true when p and q match: T↔T≡T, F↔F≡T
Negation of a statement
- negation ~p is true when p is false
- ~p is false when p is true
- negation of negation returns the original: ~(~p) = p
Statement Patterns
- a proper combination of letters and connectives is a statement pattern
- also called a proposition, e.g. p→q, p∧q, ~p∨q
- the letters used, here p and q, are its prime components
- the truth table lists the pattern's value for every combination of components
- n prime components need 2^n rows
Logical Equivalence
- A and B are logically equivalent (A≡B) when truth tables match
- match means identical, row for row
- fastest check: build both truth tables under one set of p, q, r columns
- then compare the final columns directly
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Mathematical Logic sheet covers
- Statements and Logical Connectives
- Truth Tables
- Conjunction, p∧q
- Disjunction, p∨q
- Conditional, p→q
- Biconditional, p↔q
- Connectives as Logic Gate Symbols
- Negation of a statement
- Statement Patterns
- Logical Equivalence
- Tautology, Contradiction, Contingency
- Tautology, t
- Contradiction, c
- Contingency
- Quantifiers
- Negation
- Negation of compound statements
- Converse, Inverse, Contrapositive of p→q
- Converse
- Inverse
- Contrapositive
- Algebra of statements
- idempotent
- commutative
- associative
- distributive
- De Morgan's
- identity
- complement
- absorption
- conditional law
- biconditional law
- Special cases: conditional and quantifiers
- Duals
- Symbols
- Series and Parallel Switches
- Series, S1 and S2
- Parallel, S1 and S2
- Worked: proving a logical equivalence
- Equivalent circuits
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Mathematical Logic the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.