Discrete Math Proof: Proving Equivalence of 4 Statements

In summary: You cannot just add exponents like that. We are not dealing with a power of 10 here!In summary, the four statements are equivalent because they all involve the concept of odd and even numbers in different forms. Through various proof techniques, it can be shown that if one statement is true, then the others must also be true. However, it is important to understand the definition of odd and even numbers, as well as the use of proof techniques, in order to successfully prove the equivalence of these statements.
  • #1
MarcL
170
2

Homework Statement


Prove that the following four statements are equivalent:
(a) n2 is odd.
(b) 1 − n is even.
(c) n2 is odd.
(d) n2 + 1 is even.

Homework Equations



None really, just the use of different proofs ( indirect, etc...)

The Attempt at a Solution



I'm having trouble with this one because of (2) things. First, the question makes no sense ( unless I'm reading it wrong) because, as I understand it now, it is asking to prove all statements are equivalent BUT 2 of them are eve and the rest is odd. Secondly, I'm having trouble with it because of the definition that n=2k is EVEN and n=2k+1 is odd because I tried this:

Supposle n is odd.
n=2k therefore n=(2k)2= 22k2 = 2(2k2) which goes against the definition.

Can anyone point me in the right direction?
 
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  • #2
MarcL said:
(a) n2 is odd.
(b) 1 − n is even.
(c) n2 is odd.
(d) n2 + 1 is even.
Uh ... are you sure you copied (a) and (c) correctly?
 
  • #3
I promise you I did, if you want to see the assignment yourself you can check :P --> #4 http://users.encs.concordia.ca/~grahne/comp232/assgn2.pdf
 
  • #4
MarcL said:
I promise you I did, if you want to see the assignment yourself you can check :P --> #4 http://users.encs.concordia.ca/~grahne/comp232/assgn2.pdf
Exactly. You copied it incorrectly, as is obvious.
 
  • #5
not at all, my assumption of n being even was just wrong, but that's all I've been taught before so I thought it was a set definition. anyway case closed.
 
  • #6
MarcL said:
not at all, my assumption of n being even was just wrong, but that's all I've been taught before so I thought it was a set definition. anyway case closed.
You misunderstand. You copied the problem incorrectly. Do you not see the obvious mistake?
 
  • #7
I somehow read d... definitely sorry about that. I'll go crawl in a hole now.
 
  • #8
" I'll go crawl in a hole now." Can't see any reason to do that! Just change c) to n3 is odd , then carry on. No need to worry about a trivial typo.

So, " n=2k therefore n=(2k)2 " what makes you say that?
It's just like saying n=n2 which is not generally true.
 

1. What is discrete math proof?

Discrete math proof is a method used in mathematics to prove the equivalence of different statements or theorems. It involves breaking down a complex problem into smaller, more manageable parts and using logical reasoning and mathematical principles to show that each part is equivalent to the others.

2. Why is it important to prove equivalence of statements in discrete math?

Proving equivalence of statements in discrete math is important because it allows us to establish a deeper understanding of the relationships between different mathematical concepts. It also helps to verify the accuracy and validity of these concepts, which is crucial for building a strong foundation in mathematics.

3. What are the four statements that are commonly proven to be equivalent in discrete math?

The four statements that are commonly proven to be equivalent in discrete math are the conditional statement, the contrapositive statement, the converse statement, and the biconditional statement. These statements are equivalent when they have the same truth values in all cases.

4. What is the process for proving the equivalence of four statements in discrete math?

The process for proving the equivalence of four statements in discrete math involves breaking down each statement into smaller parts, analyzing the logical relationships between these parts, and using logical reasoning to show that the statements are equivalent. This often involves using proof techniques such as mathematical induction, direct proof, and proof by contradiction.

5. What are some common challenges in proving equivalence of statements in discrete math?

Some common challenges in proving equivalence of statements in discrete math include identifying the correct proof technique to use, ensuring that all steps in the proof are valid and logical, and being able to think abstractly and conceptually to see the connections between different parts of the statements. It also requires a strong understanding of mathematical principles and techniques.

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