Mathematics 243, section 1 – Algebraic Structures Practice Questions for Exam 1

Note: This list is slightly longer than the actual exam will be to show the range of different kinds of questions that might appear.

I. Let

U = Z
A = {x ∈ Z : x is even and − 3 ≤ x ≤ 10}
B = {1, 2, 3, 4, 5}
C = {x ∈ Z : x < −4 or x > 7}

A) What is the set A ∩ (B ∪ C). B) What is the set C′?

II. Let A, B be any two subsets of a universal set U. Prove that (A ∩ B)′ = A′ ∪ B′ (where X′ is the complement of X).

III. Consider the following statement, where A is a nonempty set. “If f : A → B is an onto mapping, then for all b ∈ B, the set f^{−1}({b}) has at least one element.” A) Give the contrapositive form of this statement. B) Give the converse of the statement. Is the converse true or false? Explain.

IV. Let f, g : Z → Z be the mappings

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

A) Let S = {1, 2, 3} and T = {0, 3, 6}. What is g^{−1}(T) ∪ f(S)? B) What is the mapping f ∘ g? C) Prove that the mapping f is a permutation of Z. D) Is the mapping g a permutation of Z? Prove your assertion.

V. Let ∗ be the binary operation on R defined by x ∗ y = 3xy + x + y. A) Prove that ∗ is commutative. B) Is ∗ associative? Prove your assertion. C) Does ∗ have an identity element? Prove your assertion.

VI. A) Let M = M_{m×n}(R) be the set of all m × n matrices with real entries. Show that the

operation of matrix addition is a commutative and associative binary operation on M.

B) Let O_{n×p} be the n × p matrix all of whose entries are zero. Show that if A is any m × n matrix, then AO_{n×p} = O_{m×p} (the m × p zero matrix).