By Kenneth B Stolarsky
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2) in the second half of the 19th century. Today set theory is a fully developed, fundamental and very important branch of mathematics. Because sets are such basic mathematical objects, no precise and formal explicit mathematical definition of a set can be given, but only an intuitive idea (which everybody has to a certain extent). Of course we could try to define a set as a collection of objects, but then we would have to define what we mean by a collection, and that definition would again involve terms that need to Logic and Discrete Mathematics: A Concise Introduction, First Edition.
5. Equivalence relations and partitions . . . . . . . . . . . . . 6. Ordered sets . . . . . . . . . . . . . . . . . . . . 7. An introduction to cardinality . . . . . . . . . . . . . . . 8. Isomorphisms of ordered sets. Ordinal numbers . . . . . . . . . . 9. Application: relational databases . . . . . . . . . . . . . . 80 Set theory is one of the most fundamental parts of modern mathematics. Every mathematical object you are likely to encounter is built (or can be built) from sets using set theoretic operations.
A × (B ∪ C) = (A × B) ∪ (A × C), (B ∪ C) × A = (B × A) ∪ (C × A) 2. A × (B ∩ C) = (A × B) ∩ (A × C), (B ∩ C) × A = (B × A) ∩ (C × A) 3. A × (B − C) = (A × B) − (A × C) Proof: We prove the identity 1. For the inclusion A × (B ∪ C) ⊆ (A × B) ∪ (A × C), suppose that x ∈ A × (B ∪ C). Then x = (y‚ z) for some y ∈ A and z ∈ (B ∪ C). Since z ∈ (B ∪ C), we have z ∈ B or z ∈ C. Therefore (y‚ z) ∈ A × B or (y‚ z) ∈ A × C, and hence x = (y‚ z) ∈ (A × B) ∪ (A × C). For the inclusion (A × B) ∪ (A × C) ⊆ A × (B ∪ C) suppose that x = (y‚ z) ∈ (A × B) ∪ (A × C).