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Sets and operations

Membership, subsets, union, intersection, complement, and Venn diagrams.

A set is a collection; the operations combine them: union (either), intersection (both), difference, complement. Picture it: a Venn diagram; each region is a combination of memberships. Think it: the operations satisfy the same laws as and/or/not, sets and logic are the same algebra (Boolean algebra).

Kushanda muenzaniso · {1,2,3,4} union {3,4,5,6}

Set-Theory · Sets

{1,2,3,4} union {3,4,5,6}

Step by step

  1. A = \left\{1, 2, 3, 4\right\},\quad B = \left\{3, 4, 5, 6\right\}

    The two sets.

  2. A \cup B = \left\{1, 2, 3, 4, 5, 6\right\}

    Keep everything in either set.

  3. |A| = 4,\ |B| = 4,\ |A \cup B| = 6

    Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.

Kuratidza mhinduro
A \cup B = \left\{1, 2, 3, 4, 5, 6\right\}

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Maitiro: Sets and operations

  1. Write each set explicitly, or describe it by a property.
  2. For a union keep everything in either; for an intersection only what is in both.
  3. Check |A ∪ B| = |A| + |B| − |A ∩ B|.

Mabvunzo vanhu vanobvunza

Are some infinities bigger than others?

Yes. The integers and the rationals can be listed; the real numbers cannot (Cantor's diagonal argument), so there are strictly more reals than integers.

What is the difference between a relation and a function?

A relation pairs inputs with outputs freely; a function is a relation in which every input gets exactly one output.

More in Set Theory & Logic

Relations, functions and equivalenceCardinality and infinityLogic and methods of proof