A #~{mapping} _ &theta. #: A -> B _ is a rule which assigns to each element _ ~x _ of the set A an element _ ~xθ _ of the set B.
If _ &theta. #: A -> B _ and _ &phi. #: C -> D _ are mappings, then they are said to be #~{equal} if _ A = C , _ B = D, _ and _ ~x&theta. = ~x&phi. _ for every ~x &in. A.
The #~{identity mapping} on A is the mapping _ &iota._A #: A -> A , _ such that _ ~x&iota._A = ~x , _ for every ~x &in. A.
If _ &theta. #: A -> B _ and _ &phi. #: B -> C , _ then the #~{composite} of &theta. and &phi. is the mapping _ &theta.&phi. #: A -> C , _ such that _ ~x(&theta.&phi.) = (~x&theta.)&phi..
1) _ If _ &theta. #: &reals. -> &reals. , _ ~x&theta. = ~x^2 , _ and _ &phi. #: &reals. -> &reals. ,# _ ~x&phi. = ~x + 3 , _ then _ ~x(&theta.&phi.) = ~x^2 + 3 , _ and _ ~x(&phi.&theta.) = ( ~x + 3 )^2.
2) _ &theta. #: \{ 1, 2, 3, 4 \} -> \{ 1, 2, 3 \} , _ 1&theta. = 2 , _ 2&theta. = 3 _ 3&theta. = 2 _ 4&theta. = 1 , _ and
_ _ &phi. #: \{ 1, 2, 3 \} -> \{ 1, 2, 3, 4 \} , _ 1&phi. = 2 , _ 2&phi. = 3 , _ 3&phi. = 4 , _ then
&theta.&phi. #: \{ 1, 2, 3, 4 \} -> \{ 1, 2, 3, 4 \} , _ _ 1&theta.&phi. = 3 , _ 2&theta.&phi. = 4 , _ 3&theta.&phi. = 3 , _ 4&theta.&phi. = 2 .
The operation of composition is associative, i.e. if _ &theta. #: A -> B , _ &phi. #: B -> C , _ and _ &rho. #: C -> D , _ then _ &theta.(&phi.&rho.) _ = _ (&theta.&phi.)&rho..
Proof: _ If _ ~x &in. A , _ ~x(&theta.(&phi.&rho.)) _ = _ ~x&theta.(&phi.&rho.) _ = _ ((~x&theta.)&phi.)&rho. _ = _ (~x(&theta.&phi.))&rho. _ = _ ~x((&theta.&phi.)&rho.).
If _ &theta. #: A -> B _ is a mapping, then _ &iota._A&theta. _ = _ &theta. &iota._B.
&theta. #: A -> B _ then its #~{image}, _ Im &theta. _ #:= _ \{ ~x&theta. | ~x &in. A\} .
In example 2 above, _ Im &theta. _ = _ \{ 1, 2, 3 \} , _ _ Im &phi. _ = _ \{ 2, 3, 4 \}
Note that _ Im &theta. _ &subseteq. _ B .
&theta. #: A -> B _ is said to be #~{injective} or #~{one-to-one} if, for any ~x, ~y &in. A , _ ~x&theta. = ~y&theta. _ => _ ~x = ~y. _ This means that distinct elements in A are mapped to distinct elements in B. [ i.e. _ ~x != ~y _ => _ ~x&theta. != ~y&theta. ].
#{Theorem:} _ &theta. #: A -> B injective _ _ <=> _ _ &exist. _ &phi. #: B -> A _ such that _ &theta.&phi. = &iota._A .
Proof: _ Suppose &theta. injective, then if _ ~b &in. Im &theta. _ then &exist. unique ~a &in. A , _ ~a&theta. = ~b , _ put _ ~b&phi. = ~a . _ If _ ~b &nin. Im &theta. , _ put _ ~b&phi. = ~a_1 , _ where ~a_1 is a arbitrary element of A. _ So we have _ ~a&theta.&phi. = ~b&phi. = ~a.
Conversely, _ if _ &exist. &phi. #: B -> A _ such that _ &theta.&phi. = &iota._A , and ~x, ~y &in. A, such that _ ~x&theta. = ~y&theta. , _ then
_ _ _ ~x _ = _ ~x&iota._A _ = _ ~x&theta.&phi. _ = _ ~y&theta.&phi. _ = _ ~y&iota. _ = _ ~y
&theta. #: A -> B _ is said to be #~{surjective} or #~{onto} if _ Im &theta. = B
#{Theorem:} _ &theta. #: A -> B surjective _ _ <=> _ _ &exist. _ &phi. #: B -> A _ such that _ &phi.&theta. = &iota._B .
Proof: _ Suppose &theta. surjective, then for any ~b &in. B , &exist. at least one ~a &in. A _ such that _ ~a&theta. = ~b , _ put ~b&phi. = ~a , _ some ~a for which ~a&theta. = ~b. . _ Obviously _ ~b&phi.&theta. = ~b.
Conversely, _ if _ &exist. &phi. #: B -> A _ such that _ &phi.&theta. = &iota._B , then for any ~b &in. B, _ ~b = ~b&phi.&theta. = (~b&phi.)&theta. , _ i.e. ~b &in. Im &theta. .
&theta. #: A -> B _ is said to be #~{bijective} or a #~{one-to-one correspondence} if it is both injective and surjective
#{Theorem:} _ &theta. #: A -> B surjective _ _ <=> _ _ &exist. mapping _ &phi. #: B -> A _ such that _ &theta.&phi. = &iota._A _ and _ &phi.&theta. = &iota._B
Proof: _ By the theorems for injective and surjective mappings, &exist. _ &phi._1 #: B -> A _ such that _ &theta.&phi._1 = &iota._A , _ and _ &phi._2 #: B -> A _ such that _ &phi._2&theta. = &iota._B , _ then
_ _ _ _ &phi._1 _ = _ &iota._B&phi._1 _ = _ (&phi._2&theta.)&phi._1 _ = _ &phi._2(&theta.&phi._1) _ = _ &phi._2&iota._A _ = _ &phi._2
The converse follows immediately from the results for injective and surjective mappings.
It follows from the above proof that there is just one mapping _ &phi. #: B -> A _ such that _ &theta.&phi. = &iota._A _ and _ &phi.&theta. = &iota._B . _ This mapping is called the #~{inverse} of &theta. , and is denoted _ &theta.^{-1} .
Note that: _ _ _ ( &theta.^{-1} )^{-1} _ = _ &theta. .
#{Example}
Suppose _ _ &theta. #: &reals. -> &reals. , _ ~x&theta. = fract{5~x,~x - 2} , _ ~x != 2 ; _ 2&theta. = 5 .
Then _ _ _ _ &theta.^{-1} #: &reals. -> &reals. , _ ~y&theta.^{-1} = fract{2~y,5 - ~y} , _ ~y != 5; _ 5&theta.^{-1} = 2 .
If _ &theta. #: A -> B _ and _ &rho. #: B -> C , _ then:
Proof:
| A | zDgrmRight{&theta.,}&br.zDgrmLeft{,&phi.} | B | zDgrmRight{&rho.,}&br.zDgrmLeft{,&psi.} | C |