Project Part 2. Aaeron Jhonson-Whyte • Akuang Saechao • Allen Saeturn. Section 4.6, Problem 28. Prove the statement in two ways: (A) by contraposition and (B) by contradiction. For all integers m and n , if mn is even then m is even or n is even.
Aaeron Jhonson-Whyte • Akuang Saechao • Allen Saeturn
For all integers m and n, if mn is even then m is even or n is even.
We have to take the contrapositive of the statement by using the definition of contraposition,
For all integers m and n, if m is odd and n is odd then mn is odd.
Hence, original statement is true by contraposition.
Make the statement false by method of contradiction,
For some integers m and n, mn is even, and, m is odd and n is odd.
If we do exactly as part (a) then we can see that the contradiction will never prove true because two odd numbers multiplied will be odd. Hence original statement is true by contradiction.
6 + 15 + 20 + 15 + 6 + 1 = 63symbols can be represented in the Braille code.
19 a. Let A =
Find A2and A3.
Let G be the graph with vertices V1, V2, and V3 and with A as its adjacency matrix. Use the answers to part (a) to find the number of walks of length 2 from V1 to V3 and the number of walks of length 3 from V1 to V3. Don’t draw G to solve this problem.
To find the walks of length n from Vi to Vj,you have to look at the ijth element of An. If A is the adjacency matrix of graph G, the ijth element of Anis equal to the number of walks of length n connecting vertices Vi and Vj.
Therefore, the number of walks of length 2 from V1toV3is 3, and the number of walks of length 3 from V1 toV3 is 15.
Examine the calculations you performed in answering part (a) to find 5 walks of length 2 from V3toV3. Then draw G and find the walks by visual inspection.
Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen, France. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli. Pascal also wrote in defense of the scientific method. While he was a teenager, he worked on calculation machines, being on of the two inventors of the mechanical calculator. Pascal continued to influence mathematics throughout his life. He created the Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle") of 1653 described a convenient presentation for binomial coefficients, renamed as Pascal's triangle. Pascal also contributed philosophies to mathematics, work to physics, religion, and literature. He lived from 19 June 1623 – 19 August 1662.
Method of Proof by Contradiction:
1. Suppose the statement to be proved is false. That is, suppose that the negation of the statement is true.
2. Show that this supposition leads logically to a contradiction.
3. Conclude that the statement to be proved is true.
Definition of Contraposition:
The contrapositive of a conditional statement of the form “If p then q” is: If ∼q then ∼p.
Definition of Difference:
The difference of B minus A (or relative complement of A in B), denoted B− A,is the set of all elements that are in B and not A.
Definition of Odd and Even:
An integer n is even if, and only if, n equals twice some integer. An integer n is odd
if, and only if, n equals twice some integer plus 1.
Symbolically, if n is an integer, then
n is even ⇔ ∃ an integer k such that n=2k.
n is odd ⇔ ∃ an integer k such that n=2k+1
Definition of Union:
The union of A and B, denoted A∪B, is the set of all elements that are in at least one of A or B.
Definition of Intersection:
The intersection of A and B, denoted A ∩ B, is the set of all elements that are common to both A and B.