Mathematics 243, section 1 – Algebraic Structures Midterm Exam 1 – September 29, 2006

Directions

Do all work in the blue exam booklet. There are 100 possible points.

I. For this problem,

U = Z
A = {x ∈ Z : x is odd and − 10 ≤ x ≤ −4}
B = {−2, −1, 0, 1}
C = {x ∈ Z : x < 0}

and f : Z → Z is the mapping defined by

f(x) = \{ \begin{matrix}x + 3 & if x is even \\ x − 1 & if x is odd \end{matrix} .

A) (10) What is the set A ∪ (B ∩ C′)? B) (10) What is the set f(B)? C) (10) Given: f is a permutation of Z (you don’t need to show this). Find the inverse

mapping f^{−1} : Z → Z.

II. Let f : A → B be any mapping. A) (15) Prove that for any subsets T_{1} and T_{2} of B, f^{−1}(T_{1} ∪ T_{2}) = f^{−1}(T_{1}) ∪ f^{−1}(T_{2}). B) (10) Is f^{−1}(f(S)) always equal to S for all subsets S ⊆ A? If you say no, give a

counterexample; if you say yes, say why.

III. Let ∗ be the binary operation on R defined by x ∗ y = xy − 2x. A) (10) Show with specific counterexamples that ∗ is not associative, and not commuta-

tive. B) (15) An element e is said to be a left identity for a binary operation if e ∗ x = x for

all x. Similarly, e is a right identity if x ∗ e = x for all x. Does the ∗ operation have a left identity? Does it have a right identity? If so, say what e is in each case. If not, explain how you reach your conclusions.

IV. In both parts of this problem,

A = ( \begin{matrix}1 & 1 \\ 3 & 4 \end{matrix} ) B = ( \begin{matrix}4 & −1 \\ −3 & 1 \end{matrix} ) .

A) (10) Compute the matrix product AB or say why it doesn’t exist. B) (10) Define a mapping

F : M_{2×2}(R) → M_{2×2}(R)

for X a 2 × 2 matrix by by F (X) = BX (matrix product). Prove that F is one-to-one (injective) and give its left inverse mapping.