Prove the second absorption law by showing that if A and B are sets, then A ∩ (A ∪ B) = A.

Assume that A is a subset of some underlying universal set U. Prove the complement laws by showing that a) A ∪ A̅ = U. b) A ∩ A̅ = ∅.

Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find a) A ∪ B. b) A ∩ B. c) A − B. d) B − A.

Prove the first distributive law by showing that if A, B, and C are sets, then A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Prove the second associative law by showing that if A, B, and C are sets, then A ∩ (B ∩ C) = (A ∩ B) ∩ C.

Construct a truth table to prove each law.

$$(p\vee q)\wedge p\equiv p, \quad$$

the second Absorption Law.

Prove the absorption laws for sets.

Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find a) A ∪ B. b) A ∩ B. c) A − B. d) B − A.

Assume that A is a subset of some underlying universal set U. Prove the complementation law by showing that A̿ = A.

Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B. a) the set of sophomores taking discrete mathematics in your school. b) the set of sophomores at your school who are not taking discrete mathematics. c) the set of students at your school who either are sophomores or are taking discrete mathematics. d) the set of students at your school who either are not sophomores or are not taking discrete mathematics.

Show that the sequence {a$\square$} is a solution of the recurrence relation $a_n = a_{n−1} + 2a_{n−2} + 2n − 9$ if a) $a_n = −n + 2$. b) $a_n = 5(−1)^n − n + 2$. c) $a_n = 3(−1)^n + 2^n − n + 2$. d) $a_n = 7 \cdot 2^n − n + 2.$

Assume that A is a subset of some underlying universal set U. Prove the identity laws by showing that a) A ∪ ∅ = A. b) A ∩ U = A.

Let A be the set of students who live within one mile of school and let B be the set of students who walk to classes. Describe the students in each of these sets. a) A ∩ B b) A ∪ B c) A − B d) B − A

Assume that A is a subset of some underlying universal set U. Prove the domination laws by showing that a) A ∪ U = U. b) A ∩ ∅ = ∅.