AGRY 399 / MATH 199: Discrete Mathematics Final Exam – 20 Key Definitions & Proofs (2025 Verified Solutions)

AGRY 399 / MATH 199: Discrete
Mathematics Final Exam – 20 Key
Definitions & Proofs (2025 Verified
Solutions)
Onto Function
F is onto if and only if for all y that is an element Y, there exists an
element x in X such that f(x) = y.
One-to-One Function
A function from the set X to the set Y is one-to-one provided that if
f(x1) = f(x2), then x1 = x2.
Direct Image

Let f: X--> Y and A⊆X. The direct image of A, under f, is the set f(A)
= {y∈Y | there exists a∈A s.t. f(A) = Y}.
Inverse Image

Let f: X-->Y and V⊆Y. The inverse image of V under f is the set: f^-
1(V)={a∈X | f(a)∈V}.
Reflexive

R is reflexive if and only if for all a∈X, aRa
Symmetric

R is symmetric if and only if for all a, b∈X, if aRb, then bRa.
Transitive
R is transitive if and only if for all a, b, and c that are elements of X,
if aRb and bRc, then aRc.
Equivalence Relation
R is an equivalence relation if and only if R is reflexive, symmetric,
and transitive.
Equivalence Class

Let ~ be an equivalence relation on set X and a∈X. The equivalence
class of a is the set [a] = {x∈X | x ~ a}.