Binary Logic: True and False Statements
Reasoning · Logical Reasoning
“Binary” means two. In binary logic every statement can have only two values — True (1) or False (0). No third value, and never both at once. This one simple rule runs everything from truth-teller puzzles and “exactly so many statements are true” questions to the tables of AND / OR / IF-THEN, and the 0/1 of computers and logic gates.
Many state-level recruitment syllabuses list under Logical Capability: “Binary Logic, Syllogism, Clocks & Calendars, Assumptions, Blood Relations, Family Tree, Logic-based puzzles”. SSC CGL/CHSL, RRB, bank and state exams draw their truth-teller, truth-of-statements, conditional-sentence and code questions from it. This chapter covers that whole part from start to finish.
| Part | What you learn | Main tool |
|---|---|---|
| A. Foundation | What a statement is; true = 1, false = 0 | Test each sentence: is it a claim? |
| B. Truth-tellers and liars | One to five speakers, chains, “so many of us lie” | Assume, strike off on a clash |
| C. A fixed number true | Who did it, which box, number clues, contradiction checks | Count the true statements on each possibility |
| D. Connectives and tables | NOT, AND, OR, XOR, IF-THEN, IFF; negation; contrapositive | Truth table — all rows |
| E. Conditional sentences | What follows and what does not; necessary / sufficient | The four forms of “if P then Q” |
| F. 0/1 and gates | Bits, logic gates, circuits, switch-lamp | Gate table, series = AND, parallel = OR |
| G. Yes/no and weekdays | Questions to a stranger, answer words, lying days | List every hypothesis |
| H. Traps and exam style | The commonest mistakes, mixed questions | Spot the type first |
How to use this chapter
- Read — each part first explains the words, then gives the method step by step.
- Study the solved example — every step says what is done and why.
- Try it yourself — solve first, then open the solution.
- Check — every answer was computed by trying all True/False combinations; each question has exactly one answer.
A. Foundation: true or false
What a statement is, which sentences are not statements, and writing true/false as 1/0.
What is a statement
The definition of a statement, the “nothing in between, never both” law, and sentences that are not statements.
A statement is a sentence that makes a claim and about which it can be said for certain that it is true or false. “Jaipur is the capital of Rajasthan” — a claim, and correct: a true statement. “A year has 13 months” — a claim, but wrong: a false statement. A false statement is still a statement; it is just a wrong claim.
Two firm rules are hidden here. (1) Nothing in between — if a statement is not true it is false; there is no third value like “half true” or “don’t know”. (2) Never both — a statement cannot be true and false at the same time. In logic these are the law of excluded middle and the law of non-contradiction.
| Kind of sentence | A statement? | Why |
|---|---|---|
| A correct fact — “The Sun rises in the east.” | Yes (true) | A claim, and correct |
| A wrong claim — “The Moon is bigger than the Earth.” | Yes (false) | A claim, but wrong |
| About the future — “It will rain tomorrow.” | Yes | Unknown today, yet tomorrow it will be one or the other |
| A question — “What is the capital of Rajasthan?” | No | It asks, it does not claim |
| A command or request — “Please close the window.” | No | It orders, it does not claim |
| An exclamation — “What a lovely morning!” | No | A feeling, not a claim |
| An open sentence — “x + 4 = 9” or “n is even” | Not yet | Without a value for the unknown x or n it is neither; once a value is given it becomes a statement |
| A sentence calling itself false — “This sentence is false.” | No (a paradox) | Taken as true it turns false, taken as false it turns true — no value can stay |
- 1Ask: does it make a claim that can be called certainly true or false?
- 2It makes a claim but the claim is wrong — a false statement (still a statement).
- 1Ask: does it make a claim that can be called certainly true or false?
- 2It has an unknown letter (x, y, n, z); it becomes true or false only after a value is given — not yet. Such a sentence is called an open sentence.
True = 1, false = 0
The truth value of a statement, turning true/false into a 1/0 code, counting true statements and testing joined statements.
The truth value of a statement is its being true or false. Like a computer we write it as a number: true = 1, false = 0. That is why this is called binary (0/1) logic. When several statements are given, test each one separately and write its 1 or 0; then work out what the question asks — a count, a code or a joined statement.
- Statements about numbers: even/odd, multiple, prime, perfect square, digit sum — always test by calculation, not by guess. (1 is not prime; 2 is the only even prime.)
- Code: if four statements have the values 1, 0, 0, 1 the code is 1001 (in the order of the statements).
- Joined statements: “A and B” is true only when both are true; “A or B” when at least one is true; “if A then B” is false only when A is true and B false. The full tables are in part D.
- 1(1) 85 is an odd number → true → 1
- 2(2) the digits of 38 add up to 3 → false → 0
- 3(3) 63 is an odd number → true → 1
- 4(4) the digits of 13 add up to 11 → false → 0
- 5Written in order the code is 1010.
- 1Test each part first (T = true, F = false), then apply the rule of the joining word:
- 2(1) AND: F, T → F
- 3(2) OR: T, F → T
- 4(3) IF-THEN: T, F → F
- 5(4) XOR: T, T → F
B. Truth-tellers and liars
Puzzles with one to five speakers, “A says B is a liar” chains and “so many of us are liars” statements.
The rules and one speaker
Definitions of truth-teller and liar, what a liar’s statement really means, deducing facts from one speaker and the “I am a liar” paradox.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Two speakers
The four possible splits of two persons, what one statement settles and the full method.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Three speakers
The eight splits of three persons, one assumption settling the others, and striking off on a clash.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Four and five speakers
Four- and five-person puzzles: pruning early on a clash instead of writing every split.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
“A says B is a liar” — chains
Chain and loop puzzles: a “liar” statement flips the type, a “truth-teller” statement keeps it.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
“So many of us are liars”
Counting statements: suppose L liars and match with the number of true statements.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
C. A fixed number of statements true
Puzzles whose statements are about a hidden fact and where the number of true statements is given.
Two methods: assume and eliminate
Both methods for “exactly so many statements are true” puzzles, spotting a contradiction and where to begin.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Who did it? Which box?
Theft-and-mischief and prize-box puzzles: finding the culprit or box by counting true statements.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Statements about the statements
Statements in a list tell how many of the list’s own statements are true or false; solved by supposing F statements are false.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Number clues and checking for contradiction
Statements about a number thought of: exactly k true, which two can never be true together, at most how many can be true at once.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
D. Connectives and truth tables
The rules of the words that join statements (truth tables), the value of an expression, negation and contrapositive.
Six connectives: NOT to IFF
The truth tables of NOT, AND, OR (inclusive), XOR, IF-THEN and IFF, and “why is the sentence true when the premise is false”.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Value of an expression, tautology, equivalence
The truth table of a longer expression, counting rows, always-true / always-false and equivalent expressions.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Negation: forming the true opposite
The correct negation of a statement: and/or swapping, negating a condition, negating “all/some/none” and negating statements about numbers.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Converse, inverse, contrapositive
The converse, inverse and contrapositive of a conditional statement and which of them are equivalent.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
E. Conditional sentences
“If … then …” sentences and certain conclusions, “only if”, “unless”, necessary and sufficient, and chains.
A condition and a fact: what follows
What is certain once the value of P or Q is known along with “if P then Q”, and chain conclusions.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Only if · unless · necessary and sufficient
Recognising sentences with “only if”, “unless”, “necessary” and “sufficient” as conditionals.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
F. 0/1 codes and logic gates
The 0/1 of computers, the tables of logic gates, small circuits and switch-lamp puzzles.
Bits and logic gates
Bits (0/1), digit-by-digit operations on bit strings, binary numbers and the table of the gates.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
A small circuit: inside out
Finding the output of a circuit of two or three gates, counting input combinations and replacing a circuit by one gate.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Switches and lamps
When the lamp glows in series and parallel switch circuits, and puzzles of switches that flip with every press.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
G. Yes/no and weekday puzzles
Yes/no questions to truth-tellers and liars, choosing the right question, answer words of unknown meaning, and creatures that lie on some days of the week.
Yes/no and the right question
What the “yes/no” of a truth-teller and a liar mean, questions about oneself and the “if I asked, would you say yes” question.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Two strangers and unknown answer words
Asking two persons yes/no about each other, and an island where the words for “yes/no” are unknown.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Lying on some days of the week
A creature that lies on some days and tells the truth on the others: when can a statement be made.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
H. Traps and exam mistakes
The commonest traps of binary logic: the liar’s “everything opposite”, “I am a liar”, inclusive versus exclusive OR, wrong reversed conclusions.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Summary table and strategy
Every question type, its first move and its key fact in one place; plus one general order for all.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Mixed practice
Questions of all parts in three levels — the type is not announced, as in the exam.
Basic practice
Spotting a statement, two or three speakers, truth tables, contrapositive, gates and yes/no.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Standard practice
Counting puzzles, who did it, negation, “only if”, circuits, switches and yes/no.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Advanced practice
Loop chains, elimination in exactly-one-true, counting statements, three-variable expressions, chains and weekday puzzles.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.
Revision — cards
Think of the answer yourself first, then flip.
This part is in All Access.
With the pass, every example, practice question and solution in this chapter opens.