UNIT 2 PART 4

Question 1

What is a function?

Answer 1

two sets A and B
f:A->B IF FOR EVERY ELEMENT IN A -> image in B
one cannot branch out into two

Question 2

domain and codomain

Answer 2

f:A->B
A=DOMAIN
B=CODOMAIN

Question 3

Types of functions

Answer 3

identity: f(a) = a
constant: f(a) = c
onto,surjective: every image in B, pre-image in A
one-one,injective:diff ele in A,diff ele in B, two cannot branch out to one
bijection:one-one + onto
many to one:many ele map to one

Question 4

composition of function

Answer 4

f:x->y f:y->z
x belongs to X, z belongs to Z, exists y in Y f(x) = y and g(y) = z
same as relation except gof = f -> g not in order

Question 5

Equivalence classes

Answer 5

R -> equivalence relation
x blong X
y blng X and xRy
.

Question 6

Inverse functions

Answer 6

~
reverse each ordered pair
not always a function, only if f was one-one

Question 7

identity mapping

Answer 7

.
f:x->y g:y->x
g is finverse if gof =Ix anf fog = Iy
(gof)inverse = finverse o g inverse

Question 8

properties of function

Answer 8

f,g in Fx -> fog and gof also in Fx (Closure property of op of comp)
f,g,h in Fx (fog)oh associative
fof-1 = Ix
foIx = f

Question 9

recursive functions

Answer 9

func specify f(x) for particular val, rest with val

Question 10

algebraic structure

Answer 10

sys = nonempty set + one or more n-ary operations
{S,f1,f2..}
CADCIII

Question 11

Homomorphism

Answer 11

same type, o and * binary op

image
g:x->y homomorph g(x1ox2) = g(x1)*g(x2)
onto ->epi
1-1 ->mono
both ->endo

Question 12

ISomorphism

Answer 12

homomorphism

y sub or equal to x

Question 13

automorphism

Answer 13

y=x isomorphism

Question 14

semigroup

Answer 14

associative

AIDIC -> SMGA

Question 15

monoid

Answer 15

associative,identity

moniod -> subgroup

Question 16

group

Answer 16

associative,identity,inverse

group-> monoid,subgroup

Question 17

albelian

Answer 17

associative,identity,inverse,comm

group-> monoid,subgroup