Introductory Mathematics: Algebra and Analysis
1.2 Subsets
Definition 1.1
Suppose that$ Aand$ Bare sets. We say$ Ais a subset of$ Bif whenever$ x \in A, then $ x\in B. We write$ A \subseteq B.
Definition 1.2
Suppose that both $ A \subseteq B and $ B \subseteq A, then we say $ Aand$ B are equal and write $ A = B.
1.4 Rationals, Reals, and Pictures
Propositions 1.1
Suppose that $ a, bare real numbers, and that $ a < b. It follows that there is a rational number $ x in the range $ a < x < band an irrational number$ yin the range $ a < y < b.
1.12 Informal Description of Maps
Definition 1.3
The set$ A(the possible inputs) is called the domain of the map. The set$ B(the set from which the outputs will be drawn) is called the codomain of the map. We say that the mapping is from$ Ato$ B.
Ex) 'Only feed me data from the set$ \mathbb{Z}; output is guranteed to be from the set$ \mathbb{N}.'
$ f:\mathbb{Z}\rightarrow\mathbb{N};\; f:x\mapsto 1+x^2 \; \forall x\in \mathbb{Z}.
Definition 1.4
Given any set$ Awe can always form the identity map from $ Ato$ A;
$ \text{Id}_A:A\rightarrow A; \text{Id}_A:a\mapsto a\; \forall a\in x.
Definition of image of$ f
Suppose that $ f:A\rightarrow Bis a map. The image of$ f is
$ \text{im}(f) = \{x | x\in B, x= f(a) \text{ for some } a\in A\}or
$ \text{im}(f) = \{f(x) | x\in A \}.
Definition of surjective
A map is onto or surjective or epic when $ \text{im}(f) = \text{cod}(f).
Definition of injective
A map$ f: A\rightarrow Bis 1-1 or injective or monic
$ a,a^{\prime}\in A \land a \neq a^{\prime} \implies f(a) \neq f(a^{\prime}).
Definition of bijective
A map is bijective or a bijection when it it both injective and surjective. It is called 1-1 correspondence as well.
Definition of equality between maps
Tow maps$ fand$ gare equal when
dom(f) = dom(g) $ \land cod(f) = cod(g) $ \land $ \forall x\in \text{dom}(f), f(x)=g(x).
Definition of composition of maps
Suppose that$ f: A\rightarrow Band$ g:B\rightarrow C, the we can define a map called$ g\circ fas
$ g\circ f: A\rightarrow C; \; g\circ f: x\mapsto g(f(x))\; \forall x\in A.
Note: we can form $ g\circ f if and only if cod(f) = dom(g).
Proposition 1.2
(a) The composition of two injective maps is injective.
(b) The composition of two surjective maps is surjective.
$ \rightarrowThe composition of two bijective maps is bijective.