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Introduction to Boolean Algebra

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Introduction to Boolean Algebra

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- George Boole
- English mathematician
- Developed a mathematical model for logical thinking

- Mathematical Analysis of Logic (1847)
- Applicability of Boolean logic to electrical circuits independently recognized in writings by
- Claude Shannon (USA, 1937)
- Victor Shestakov (Russia, 1941)

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- Definition: A mathematical model of propositional logic and set theory.
- Revisit the properties of wffs in chapter one and of sets in chapter three.
- Note the similarities.

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- “v” means disjunction.
- “^” means conjunction.
- “0” -> contradiction.
- “1” -> tautology.
- Propositional logic, set theory, and Boolean logic share common properties.

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- Similarities between propositional logic and set theory:
- Both are concerned with sets
- Set of wffs
- Sets of subsets of a set

- Both have 2 binary operations and 1 unary operation.
- Each has 2 distinct elements.
- Each has 10 properties.

- Both are concerned with sets
- These are distinguishing features of a Boolean algebra.

- Definition: A set B having
- Defined binary operations + and ∙
- Defined unary operation ‘
- Two distinct elements 0 and 1
- With the following properties for elements of B:
- Commutativity
- Associativity
- Distribution
- Identity
- Complementation

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- B ={0, 1}
- +, · and ‘ defined by truth tables, p. 538.
- The properties defining a Boolean algebra can be demonstrated to hold for B.
- Try practice 1, p. 538.

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- Note that in order to apply the properties of Boolean algebra, there must be an exact match of patterns.
- See demonstration, p. 539.

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- The mapping of elements of one instance to the elements of the other so that crucial properties are preserved.
- [Recall the definition of bijection: If every element in the domain has a unique image in the codomain and every element in the codomain has a unique preimage, then the function is a one-to-one onto function, also known as a bijective function, or a bijection.]

- See Example 5, p. 542.
- See definition, p. 544.

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- The number of elements in a Boolean algebra, B, is a power of 2 (say, 2m).
- B is isomorphic to a power set with elements from 1 to m.

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- Boolean algebras, propositional logic, and set theory are abstractions of common properties.
- If an isomorphism exists from A to B, which are instances of a structure, then A and B are equivalent except for labels.
- All finite Boolean algebras are isomorphic to Boolean algebras that are power sets.

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