Injective relation
Bijective
relation
Ex^w^/e 2.d.7. Let ^ - the set of real numbers, ^ the set of real positive
numbers, and a function / : ^ ^ R.
If ^ = R = ^ then the function /: x ^ x2 gives the map of ^ onto R which is not surjective.
If ^ = R = ^ then the function /: x ^ 4x - 3 gives the map of ^ onto R which is surjective.
If ^ = R = ^ ^ then the function /: x ^ 3x gives the map of ^ onto R which is injective.
Consider functions / : ^ ^ R and g: R ^ C ; that is, where the range of/ is the domain of g. Then we may define a new function from ^ to C, called the composition of/ and g and written g o / as follows:
(g o / ^ g(/(a))
That is, we find the image of a under / and then find the image of /(a) under g.
Consider any function / : ^ ^ R. Then / o = / and /R o / = /, where and 7R are the identity functions on ^ and R, respectively. The mapping defined by these formulas is called identical. Thus
7.4
=
^..^w^/e 2.d.2. Let the mapping / be given by the table
then the mapping / 1 ^ 4 x R is defined by the table
The functions / and / 1we write in the form
1
2 3 4 5 4 3 5 1 2
1
2 3 4 5 4 5 2 1 3
/-1
=
/
=
Let us check the fulfillment of the conditions 74 o / = / and / o 74 = /
/4 o / =
'1 2 3 4 5' |
'1 2 3 4 5' |
|
'1 2 3 4 5' |
1 2 3 4 5_ |
4 3 5 1 2_ |
|
4 3 5 1 2_ |
= /
In the similar way we get / o 74 = / .2.8. Ordered Sets
A binary relation ^ on a set ^ is called an order relation or partial order relation if it is antisymmetric and transitive.
A binary relation ^ on a set ^ is called an nonstrict order relation if it is reflexive, antisymmetric and transitive.
A binary relation ^ on a set ^ is called a strict order relation if it is antireflexive, antisymmetric and transitive.
If an order relation is total then it is called a totally ordered or
linearly ordered.
A nonstrict order relation is denoted by " < ", and strict order relation by "< and ^ < ^ is read precedes ^ < ^ means ^ < ^ and ^ # ^, and is read strictly precedes or strictly succeeds ^
De/zM/Y/oM. Let ^ be a subset of a partially ordered set An element M in ^ is called an upper bound of ^ if M succeeds every element of i.e. if for every x in we have x < M.
Analogously, an element w in ^ is called a lower bound of a subset ^ of if w precedes every element of i.e. if for every y in we have w < y.
2.9. Suplementary Problems
Which of the following sets are equal?
^ = {x: x2 - 4x + 3 = 0j, C = {x: x e #, x < 3}, F = {l,2}, C = {3,l},
R = {x: x2 - 3x + 2 = 0} D = {x: x e #, x is odd, x < 5}, F = {l,2,l}, ^ = {l,1,3}.
Let ^ = {1,2,..., 8,9}, R = {2,4,6,8}, C = {l, 3,5,7,9}, D = {3,4,5}, F = {3,5}. Which of above sets can equal a set ^ under each of the following conditions?
^ and R are disjont. (c) ^ ^ ^ but ^ ^ C .
^ ^ D but ^ ^ R (d) ^ ^ C but ^ ^ ^ .
Let ^ = {<7, c, ^, e}, R = {<7, ^, /, g}, C = c, e, g, A}, D = {i/, e, /, g, A}. Find:
(a) ^ ^ R ^ ^(R u D) (g) u D) \ C (/) ^ e R
(A) R ^ C R \ (C u D) (A) R ^ C ^ D ^ e C
(c) C \ D (/) (^ u DR (i) (C \ \ D (/) (^ e D)\ R
Draw a Venn diagram of sets R, C where ^ ^ R, sets R and C are disjoint, but ^ and C have elenments in common.
Consider the set Q of rational numbers with the order <. Consider a subset D of Q defined by D = {x x e Q and 8 < x^ < 15}. Find the upper and lower bounds.