mathematics

Simple theorems in the algebra of sets

Simple theorems in the algebra of sets is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Simple theorems in the algebra of sets rather than just read about it. In short: The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.

Key takeaways

Simple theorems in the algebra of sets belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Simple theorems in the algebra of sets to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Simple theorems in the algebra of sets from memory before moving on to harder problems.

Reference excerpt

The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}. The algebra of sets describes the properties of all possible subsets of U, called the power set of U and denoted P(U). P(U) is assumed closed under union, intersection, and set complement. The algebra of sets is an interpretation or model of Boolean algebra, with union, intersection, set complement, U, and {} interpreting Boolean sum, product, complement, 1, and 0, respectively. The properties below are stated without proof, but can be derived from a small number of properties taken as axioms. A "*" follows the algebra of sets interpretation of Huntington's (1904) classic postulate set for Boolean algebra. These properties can be visualized with Venn diagrams. They also follow from the fact that P(U) is a Boolean lattice. The properties followed by "L" interpret the lattice axioms. Elementary discrete mathematics courses sometimes leave students with the impression that the subject matter of set theory is no more than these properties. For more about elementary set theory, see set, set theory, algebra of sets, and naive set theory. For an introduction to set theory at a higher level, see also axiomatic set theory, cardinal number, ordinal number, Cantor–Bernstein–Schroeder theorem, Cantor's diagonal argument, Cantor's first uncountability proof, Cantor's theorem, well-ordering theorem, axiom of choice, and Zorn's lemma. The properties below include a defined binary operation, relative complement, denoted by the infix operator "\". The "relative complement of A in B," denoted B \A, is defined as (A ∪B′)′ and as A′ ∩B.

PROPOSITION 1. For any U and any subset A of U:

{}′ = U; 'U'′ = {}; A \ {} = A; {} \ A = {}; A ∩ {} = {}; A ∪ {} = A; * A ∩ U = A; * A ∪ U = U; A′ ∪ A = U; * A′ ∩ A = {}; * A \ A = {}; U \ A = A′; A \ U = {}; A′′ = A; A ∩ A = A; A ∪ A = A.

PROPOSITION 2. For any sets A, B, and C:

A ∩ B = B ∩ A; * L A ∪ B = B ∪ A; * L A ∪ (A ∩ B) = A; L A ∩ (A ∪ B) = A; L (A ∪ B) \ A = B \ A; A ∩ B = {} if and only if B \ A = B; (A′ ∪ B)′ ∪ (A′ ∪ B′)′ = A; (A ∩ B) ∩ C = A ∩ (B ∩ C); L (A ∪ B) ∪ C = A ∪ (B ∪ C); L C \ (A ∩ B) = (C \ A) ∪ (C \ B); C \ (A ∪ B) = (C \ A) ∩ (C \ B); C \ (B \ A) = (C \ B) ∪(C ∩ A); (B \ A) ∩ C = (B ∩ C) \ A = B ∩ (C \ A); (B \ A) ∪ C = (B ∪ C) \ (A \ C). The distributive laws:

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C); * A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). *

PROPOSITION 3. Some properties of ⊆:

A ⊆ B if and only if A ∩ B = A; A ⊆ B if and only if A ∪ B = B; A ⊆ B if and only if B′ ⊆ A′; A ⊆ B if and only if A \ B = {}; A ∩ B ⊆ A ⊆ A ∪ B.

See also List of set identities and relations – Equalities for combinations of sets

References Edward Huntington (1904) "Sets of independent postulates for the algebra of logic," Transactions of the American Mathematical Society 5: 288-309. Whitesitt, J. E. (1961) Boolean Algebra and Its Applications. Addison-Wesley. Dover reprint, 1999.

Worked examples

Example 1 — a first encounter with Simple theorems in the algebra of sets

Start with the simplest possible case. Write down what Simple theorems in the algebra of sets claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Simple theorems in the algebra of sets before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Simple theorems in the algebra of sets ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Simple theorems in the algebra of sets

In research: Simple theorems in the algebra of sets appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Simple theorems in the algebra of sets in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Simple theorems in the algebra of sets is common in secondary-school and first-year university syllabi. It links to neighbouring topics Operations on sets, so understanding it makes those chapters shorter.
In everyday life: Look for Simple theorems in the algebra of sets outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Simple theorems in the algebra of sets” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Simple theorems in the algebra of sets in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Simple theorems in the algebra of sets means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Simple theorems in the algebra of sets out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Simple theorems in the algebra of sets in simple terms?

Why does Simple theorems in the algebra of sets matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Simple theorems in the algebra of sets?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Simple theorems in the algebra of sets.