Discrete Mathematics Answers

Consider the statement: The art show was enjoyable but the room was hot.
a) Use a variable to represent each basic statement in the given statement.
b) Use the variables and logical operators to represent the statement in symbolic form.
c) Construct the truth table.

Define a function fm: N x N ->N as follows: fm(n, k) =k if 0 ≤ n < m, and fm(n, k) =fm(n-m, k+1) otherwise. Describe in terms of a single well-known arithmetic operation what fm(n, 0) is computing.

Write following using summation
n(n-1)+(n-2)++.....+1

Let A be a countable set, and B is another set. Assume further that there exists an onto function f:A->B. Is B necessarily countable? Provide a full justification for your answer.

If X,Y, and Z are sets and |X|=|Y| and |Y|=|Z|, show that |X|=|Z|. Note that we are not assuming that the given sets are finite.

(a) Show that a subset of a countable set is countable.

Let C={A1, A2, ..., An} be a collection of finite sets that are pairwise disjoint. Further suppose that |Ai|=i. Compute |U(i=1 to n)Ai|, and write your answer in the simplest closed form possible.

Let Bn={(x, y)|0≤x≤n and 0≤y≤n}, where n is a nonnegative integer. Find U(n=0 to infinity)Bn and ∩(n=0 to infinity)Bn.

If In= (-1/(5n),1/(5n)) where n≥1 is an integer and In represents an interval on the real number line, find U(n=1 to infinity) In and ∩(n=1 to infinity) In.

Let R be an equivalence relation on Z, which has P={{±i}|i ∈ N} as its collection of equivalence classes. Describe the equivalence relation R.