Mathematics 243, section 3 – Algebraic Structures Exam 1 – September 28, 2012

Directions

Do all the problems below in the blue exam booklet. Do not place anything you want to have considered for credit on the exam sheet. There are 100 regular points and 10 Extra Credit points.

I. Let A, B, C be any subsets of a universal set U. A) (15) Show that if A ⊆ B, then A ∩ C ⊆ B ∩ C. B) (10) Is it true that A − B = A − C implies B = C? If you think the statement

is true, illustrate with a Venn diagram. If you think the statement is false, give a counterexample.

II. (20) Consider the following statement: “If f : A → B is an onto mapping with A, B nonempty, then for all b ∈ B, the set f^{−1}({b}) is nonempty.” Give the contrapositive form of this statement. Are the given statement and its contrapositive true or false? Explain.

III. Let f, g : Z → Z be the mappings given by

f(x) = \{ \begin{matrix}2x − 1 & if x is even \\ x + 1 & if x is odd \end{matrix} g(x) = \{ \begin{matrix}x & if x is even \\ x + 1 & if x is odd \end{matrix}

A) (15) Let S = {1, 2, 3} and T = {0, 3, 6}. What is g^{−1}(T) ∩ f(S)? B) (10) What is the mapping f ∘ g?

IV. Consider the binary operation on Z defined by x ∗ y = x + xy + y. A) (10) Is ∗ associative? B) (5) There is an identity element for ∗. Find it and justify your claim.

V. (15) Let A ∈ M_{2×2}(Z) be the matrix A = (a_{ij}) defined by a_{ij} = i − j, and let B ∈ M_{3×2}(Z) be the matrix B = (b_{ij}) defined by b_{ij} = i + 2j. Which of the two products AB, BA are defined? Compute the one(s) that are defined.

Extra Credit (10) Which x ∈ Z have inverses for the binary operation ∗ from question IV?