Relations & Logic: Truth Tables, Sets, Boolean & Kinship

Relations & Logic packs five discrete-math tools under one tab bar: Family, Propositions, Sets, Symbolic and Boolean. Each answers by enumeration or exact rules, never by sampling. Family turns a chain of relatives into a kinship term; Propositions builds the full truth table of a formula and classifies it Tautology, Contradiction or Satisfiable (contingent); Sets computes union, intersection, difference, symmetric difference, power sets and Cartesian products; Symbolic runs computer-algebra work from simplification to Laplace transforms; Boolean evaluates one assignment of an expression. Every tab has its own address — /math/relogic/proposition is one example — so a result can be bookmarked or shared.

Updated 2026-10-01Reviewed by CalcX Editorial Team6 sources5 min read

How to use the five tabs

Pick a tabfrom the top bar, or open its URL directly.
Family — build a chain.Click relatives in order from "Me": Father then Elder Brother reads Father's Elder Brother (Uncle). ↩ Undo drops the last step; Reset clears the chain.
Propositions — generate a truth table.Type an expression, then press Generate Truth Table for the verdict, the true-row count and the first failing row.
Sets — enter items.Separate them with commas, spaces or semicolons, or type braces such as {1, 2}; duplicates collapse. Choose Operations, Power Set or Cartesian, then Compute.
Symbolic — pick an operation.Twelve modes cover simplify, expand, factor, solve, systems, derivatives, integrals, partial fractions, Laplace transforms and sums; results copy as text or LaTeX.
Boolean — set the variables.Flip each variable the parser finds between 1 and 0, then press Evaluate.

Worked examples

Every readout below comes from the live tabs; the same input reproduces it.

Tab · input	Readout
Family · Father → Elder Brother	Father's Elder Brother (Uncle)
Propositions · `(A -> B) & A -> B`	Tautology, 4 / 4 rows true
Propositions · `A -> B`	Satisfiable (contingent), 3 / 4 rows true; counterexample A=T, B=F
Propositions · `!(A & B) <-> (!A \| !B)`	Tautology, 4 / 4 rows true
Sets · A = `1, 2, 3, 4, 5`, B = `3, 4, 5, 6, 7`	A ∪ B = {1, 2, 3, 4, 5, 6, 7}, A ∩ B = {3, 4, 5}, A △ B = {1, 2, 6, 7}, \|A × B\| = 25
Sets · power set of A = `1, 2`	\|P(A)\| = 4 (2^2): ∅, {1}, {2}, {1, 2}
Symbolic · Simplify `(x^2 - 1)/(x - 1)`	1+x
Symbolic · Factor `x^2 - 5x + 6`	*(-2+x)(-3+x)**
Symbolic · Solve `x^2 - 4 = 0`	[2,-2]
Symbolic · Defint `x^2`, 0 to 1	1/3
Symbolic · Sum `k^2`, k = 1 to 10	385
Boolean · `(A \| B) & !C` with A=1, B=0, C=0	(A \| B) & !C = TRUE (1)

The symbolic tabs print in the algebra engine's canonical order: (x + 1)^3 expands to 1+3x+3x^2+x^3 (ascending powers) and (x + 1)/(x^2 - 1) part-fractions to (-1+x)^(-1). Truth tables fill from all-false downward, which is why A -> B reports A=T, B=F. A set difference that removes everything prints A − B = {}.

What each tab computes

Family resolves against a table of 80 named combinations and echoes back any chain it cannot name, such as Husband → Husband. Propositions and Boolean share one parser: ! binds tightest, then &, ^, |, ->, <->, with -> grouping to the right; word operators (NOT, AND, NAND) and symbols ¬, ∧, ∨, ⊕, ↑, ↓ also parse. Sets treats items as text and drops duplicates; Symbolic returns closed forms that are exact under its rewrite rules.

Limits to keep in mind

Finite inputs, finite listings.: Sets drops duplicate items and treats every item as text: 1, 1, 2 is a two-element set, and 01 is a different element from 1. A power set is capped at twelve elements — Set too large for power set (max 12 elements) — and a Cartesian pair list prints the first 256 pairs even when the size line reports a larger product: two 20-item sets give |A × B| = 400 with a truncated list.
What it does not do: numeric or statistical work.: Nothing here integrates numerically, estimates a limit, fits a regression or computes a distribution. For numeric calculus and step-by-step derivations use Math Tools; for statistics, hypothesis tests and distributions use Statistics. For everyday typed arithmetic with percent, use the scientific calculator.

The eight-variable ceiling

The truth-table verdict is an enumeration result: Propositions evaluates every row — all-false first — and only then prints Tautology, Contradiction or Satisfiable. That makes the boundary sharp. A formula with more than eight variables is refused with Too many variables (max 8) instead of being approximated, so a tautology over nine variables cannot be certified here. Inside the bound the search is exhaustive — at most 256 rows — and the smallest counterexample is the first failing row; read both before quoting a verdict.

Where it fits

Checking logic exercises and set identities

De Morgan's laws pass in one pass: !(A & B) <-> (!A | !B) reads Tautology, 4 / 4 rows true, while A & !A reads Contradiction, 0 / 2 rows true. With A = 1, 2 and B = 1, 2, 3, the same panel answers A ⊆ B: Yes, B ⊆ A: No and A △ B = {3}.

Naming relatives

Turn "what do I call my mother's elder sister's son?" into three clicks: Mother → Elder Sister → Son reads Older Maternal Male Cousin. Spouse-side terms resolve too — Husband → Mother is Mother-in-law, Wife → Father is Father-in-law — and unnamed pairs such as Husband → Husband come back as the literal chain.

Boolean conditions in code

Settle a condition before rewriting it: A NAND B with A=1, B=1 reads FALSE (0), and lowercase a AND !b evaluates exactly as its uppercase form because variables are single letters, case-insensitive. The truth table then shows which rows a refactor must preserve.

References

Wikipedia, Truth table, en.wikipedia.org （访问日期：2026-10-01）— rows and truth conditions.
Wikipedia, Algebra of sets, en.wikipedia.org （访问日期：2026-10-01）— union, intersection, difference and De Morgan identities.
Wikipedia, Boolean algebra, en.wikipedia.org （访问日期：2026-10-01）— connectives and precedence.
Wikipedia, Kinship terminology, en.wikipedia.org （访问日期：2026-10-01）— English terms for relatives by blood and marriage.
Wikipedia, Computer algebra, en.wikipedia.org （访问日期：2026-10-01）— exact symbolic manipulation against numeric approximation.
Stanford Encyclopedia of Philosophy, Propositional Logic, plato.stanford.edu （访问日期：2026-10-01）— satisfiability, tautology and countermodels.