California State University – MATH 101: College
Algebra Verified Practice Questions with A+
Answers | Final Exam Guide FOR 2025/2026 (the
most recent quizzes)
Injective - ✔✔if f(x) = f(y) implies x = y.
Surjective - ✔✔if,for all b∈B, there is an a∈A with f(a)=b.
Bijective - ✔✔If f is injective and surjective
Equivalence Relation - ✔✔If the following axioms hold:
1. (Reflexivity) a ∼ a
2. (Symmetry) a ∼ b implies b ∼ a
3. (Transitivity)a∼b and b∼c implies a∼c
Equivalence Class - ✔✔The equivalence class of a is the set of elements in S that are equivalent
to a; we write [a] = {x ∈ S : x ∼ a}.
Group - ✔✔If the following axioms hold:
1. (Closure) If a,b∈G then a∗b∈G
2. (Associativity)If a,b,c∈G, then (a∗b)∗c=a∗(b∗c)
3. (Identity) There is an e∈G for which e∗a=a=a∗e for all a∈G
4. (Inverse)For every a∈G, there is an element a^−1 ∈G for which a∗a−1 =e=a−1∗a.
Order of an element - ✔✔The order of the element a ∈ G is the smallest positive integer k such
that a^k = e.
Order of a group - ✔✔Let G be a group. If G is finite, then the order of the group G is the
cardinality of G. If G is infinite, then G has infinite order
Abelian - ✔✔if xy=yx for all x y in G
Subgroup - ✔✔A subset H of a group G is a subgroup if H is a group with same operation as G.
Homomorphism - ✔✔Let (G,∗) and (G′,∗′) be two groups. A homomorphism is a map φ : G →
G′ that preserves the group operations: for a, b ∈ G,
φ(a ∗ b) = φ(a) ∗′ φ(b).
Algebra Verified Practice Questions with A+
Answers | Final Exam Guide FOR 2025/2026 (the
most recent quizzes)
Injective - ✔✔if f(x) = f(y) implies x = y.
Surjective - ✔✔if,for all b∈B, there is an a∈A with f(a)=b.
Bijective - ✔✔If f is injective and surjective
Equivalence Relation - ✔✔If the following axioms hold:
1. (Reflexivity) a ∼ a
2. (Symmetry) a ∼ b implies b ∼ a
3. (Transitivity)a∼b and b∼c implies a∼c
Equivalence Class - ✔✔The equivalence class of a is the set of elements in S that are equivalent
to a; we write [a] = {x ∈ S : x ∼ a}.
Group - ✔✔If the following axioms hold:
1. (Closure) If a,b∈G then a∗b∈G
2. (Associativity)If a,b,c∈G, then (a∗b)∗c=a∗(b∗c)
3. (Identity) There is an e∈G for which e∗a=a=a∗e for all a∈G
4. (Inverse)For every a∈G, there is an element a^−1 ∈G for which a∗a−1 =e=a−1∗a.
Order of an element - ✔✔The order of the element a ∈ G is the smallest positive integer k such
that a^k = e.
Order of a group - ✔✔Let G be a group. If G is finite, then the order of the group G is the
cardinality of G. If G is infinite, then G has infinite order
Abelian - ✔✔if xy=yx for all x y in G
Subgroup - ✔✔A subset H of a group G is a subgroup if H is a group with same operation as G.
Homomorphism - ✔✔Let (G,∗) and (G′,∗′) be two groups. A homomorphism is a map φ : G →
G′ that preserves the group operations: for a, b ∈ G,
φ(a ∗ b) = φ(a) ∗′ φ(b).
