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Section 0.1: Real and Complex Number Systems

Def: Aset is a collection of objects. If xis an object in a set S, we say xis an element of

Sand we write x∈S.

If every element of a set Ais also an element of a set B, we say Ais a subset of Band

write A⊆B.

Two sets are equal if they have exactly the same elements. Then we have

A⊆B, B ⊆A⇐⇒ A=B.

Def: The union of two sets is the set containing all elements of either set.

Ex: A={1,2}and B={2,3,4}.

A∪B={1,2,3,4}

Def: The intersection of two sets is the set containing the elements that are in both sets.

Ex: A={1,2}and B={2,3,4}.

A∩B={2}

Def: The diﬀerence of two sets is the set containing the elements in one set but not in the

other.

Ex: A={1,2}and B={2,3,4}.

A\B=A−B={1}, B \A=B−A={3,4}

Note there are two acceptable notations for set diﬀerence, A\Band A−B.

Def: The natural numbers,N, are all positive numbers with no decimals.

N={1,2,3,4, . . .}

Def: The integers,Z, are the positive and negative natural numbers and 0. Note that

N⊂Z.

Z={. . . , −3,−2,−1,0,1,2,3, . . .}

Def: The rational numbers,Q, are all numbers that can be written as a

b, a fraction, where

a, b ∈Z.

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