Discrete Mathematics Answers

Determine whether f is a function from the integers to the set of all real numbers.
Enter "Y" for yes and "N" for no.
1. f(n)=±n
2. f(n)=1/(n^2−16)
3. f(n)=√n^2+6
4. f(n)=1/(n^2+9)

Use Karnaugh map to minimize the sum of product expansion
xy'z+ xy'z'+x'yz+x'y'z+x'y'z'

Use Karnaugh map to minimize the sum of product expansion
xy'z+ xy'z'+x'yz+x'y'z+x'y'z'

WHAT IS THE CARDINALITY OF {a,{a},{a,{a}}}

. The crossing number of a graph G, written ν(G), is the fewest number of nonendpoint edge-crossings that occur over all possible drawings of G in the plane. We assume that no edge crosses itself, and that edge crossings occur only at pairs of edges (ie, no three edges can cross at one point). With these conventions, we can say that G is planar if and only if ν(G) = 0.
(a) Prove that ν(K5) = 1.
(b) Prove that ν(K6) = 3 by
• exhibiting a drawing of K6 with exactly three edge-crossings, and
• assuming that K6 can be drawn with two edge crossings, introducing new vertices at the two edgecrossings, and then using Euler’s Formula to try to obtain a contradiction.

In a a class of 80 students 53 study Art 60 study Biology 36 study art and Biology 34 study art and Chemistry 6 study Biology only and 18 study biology but not chemistry. How many students study art only and how many students study chemistry

Suppose that in the world every pair of people either
(a) likes one another,
(b) dislikes one another, or
(c) is indifferent toward one another.
Prove that in any gathering of 17 people, there is a group of three people all of whom satisfy one of conditions (a), (b) or (c).

Suppose a ϵ Z. Prove by contradiction that if a2 -2a + 7 is even, then a is even.

Use proof by contraposition to show that if x + y ≥ 2, where x and y are real numbers, then x ≥ 1 or y ≥ 1.

Suppose a ϵ Z. Prove by contradiction that if a2 -2a + 7 is even, then a is even.