truth set
C2Technical / Academic
Definition
Meaning
The set of all variable assignments that make a given logical statement or equation true.
In mathematics and logic, the collection of all solutions to a predicate, equation, or system; the set of elements for which a condition holds.
Linguistics
Semantic Notes
Purely a technical term with no figurative or colloquial use. Core to predicate logic, set theory, and related disciplines.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling conventions (e.g., 'mathematics' vs. 'math') may appear in surrounding text.
Connotations
Neutral and precise in both varieties.
Frequency
Equally low-frequency and confined to technical contexts in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The truth set of [predicate/equation][Predicate/Equation] has a truth set of [set description]Vocabulary
Synonyms
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Central in logic, discrete mathematics, and theoretical computer science courses.
Everyday
Not used.
Technical
The primary domain of use, in formal specifications and mathematical proofs.
Examples
By CEFR Level
- For the equation x + 2 = 5, the truth set is {3}.
- The truth set of the inequality y < 4, where y is a natural number, is {1, 2, 3}.
- To prove the proposition's validity, one must demonstrate that its truth set under the given interpretation is non-empty.
Learning
Memory Aids
Mnemonic
Think of a TRUE/FALSE test: the TRUTH SET is the list of all answers that would get a 'TRUE' mark.
Conceptual Metaphor
SOLVING IS FINDING A LOCATION (The truth set is the 'territory' where the statement is valid).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating as 'правда сет' or 'набор правды'. The correct conceptual translation is 'множество истинности'.
Common Mistakes
- Using 'truth set' to mean a collection of true statements (it's a collection of objects that *make* a statement true).
- Confusing with 'set of truths'.
- Using in non-mathematical contexts.
Practice
Quiz
What does the 'truth set' of a logical predicate represent?
FAQ
Frequently Asked Questions
In the context of equations and inequalities, they are synonymous. 'Truth set' is a more general logical term, while 'solution set' is more common in algebra.
Yes, if no element from the domain satisfies the condition, the truth set is the empty set (∅). This represents a false statement for all inputs.
It is fundamental to mathematical logic, set theory, discrete mathematics, and the theoretical foundations of computer science.
Often by listing its elements, using set-builder notation, or, for inequalities involving real numbers, shading a region on a number line or coordinate plane.