FIRST COURSE IN ABSTRACT ALGEBRA CERTIFICATION EVALUATION TEST COMPLETE QUESTIONS AND CORRECT ANSWERS

FIRST COURSE IN ABSTRACT ALGEBRA
CERTIFICATION EVALUATION TEST COMPLETE
QUESTIONS AND CORRECT ANSWERS

◉ Subset.
Answer: Every Element of A is an element of B: A ⊆ B


◉ Power Set.
Answer: Set of all subsets


◉ Intersection (A ∩ B).
Answer: Elements in A AND in B


◉ Union (A ∪ B).
Answer: Elements in A OR in B


◉ Cartesian Product (A × B).
Answer: {(X,Y) : X ∈ A, Y ∈ B}


◉ Division (A | B).
Answer: let a, b ∈ ℤ and b ≠ 0. Then b *divides* a if ∃k ∈ ℤ s.t. a = k⋅b

, Therefore, b is a divisor of a


◉ Relation.
Answer: between X and Y is a subset of X × Y


◉ Function.
Answer: from X to Y is a relation f between X and Y s.t. each x ∈ X
appears as the first number of *exactly one* ordered pair y = f(x)


Notation:
f: X --> Y
x |--> f(x)


◉ Domain.
Answer: all numbers in 1st set (X)


◉ Codomain.
Answer: all numbers in 2nd set (Y)


◉ Range.
Answer: numbers in Y mapped to a value in X