okay, i'll try to answer some (and i'll answer it from 7-1, in accordance to the sequence of our study)..
Let be a function (to be precise).
f is said to be one-to-one if
f is said to be onto B if the range of f is B.
if f is onto and one-to-one, then f is a bijection..
A (nonempty) set G with a binary operation, say *, is said to be a group if the following are satisfied:
i) * is associative, i.e. a*(b*c)=(a*b)*c
ii) there exists an element e in G such that for every elements g in G, e*g=g=g*e (e is called the identity element of G)
iii) for every elements g in G, there exists an inverse in G such that
note: if * is commutative, i.e. for all a,b in G, a*b=b*a, then G is said to be an abelian group.
Let H be a (nonempty) subset of G. H is said to be a subgroup of G iff H is a group under the operation of G.
Let be a map of a group G into a group G'.
is a homomorphism if for all a,b in G ..(here we note that on the LHS, the operation of G is used; while on the RHS, the operation of G' is used.. also, both operation may not be distinct..)
A group homomorphism is said to be an isomorphism if is a bijection..
A (nonempty) set R with two binary operations, + and , is said to be a ring if the following are satisfied:
i) <R,+> is an abelian group. (R is an abelian group with respect to +)
ii) is associative
iii) for all a,b,c in R, (Left Distributive Law); (Right Distributive Law)
A (nonempty) set F with two binary operations, + and , is said to be a field if the following are satisfied:
i) <F,+, > is a ring.
ii) is commutative
iii) there exists an element in F such that for all elements a in F, ( (or 1) is called the multiplicative identity element of F)
iv) for every element a in F s.t , there exists an element in F st ( is called the multiplicative inverse of a) (also note that, this time, -a denotes the additive inverse of a; and 0 is the additive identity in F)