Methods of Proof

It is an argument that establishes a true statement.

• proof

• indirect

• direct

• methods

By direct proof, we assume that p is __________ and prove that q is true.

• true

• false

• even

• odd

In indirect proof, the _______ of the implication "if p then q" is considered.

• contrapositive

• inverse

• opposite

• transpose

Which of the following is correct?

• Proof by contradiction is a direct proof method.

• An irrational number can be defined as

  $\frac{a}{b}$ where $a$ and $b$ are integers, such that $b\ne0$ .
  

• A number divisible by $k$ can be represented as $n|k$

• A proof by contraposition for $p\rightarrow q$ is done by proving $\sim p\rightarrow\sim q$ .

Using the direct proof, what can be assumed so that the statement below is true?

If an integer is odd and another integer is even, their product is even.

• The integers are both odd.

• The integers are both even.

• An integer is odd and another integer is even.

• Neither of them is odd nor even.

Using the indirect proof, what can be assumed so that the statement below is true?

If the square of an integer  $n$ is even, then  $n$ itself is even.

 

•  $n$  is odd

•  $n$  is even

•  $n$  is neither odd nor even
  

•  $n$  is both even and odd

In the statement ‘if  $k$ is a rational number, then  $k+1$ is also a rational number,’ what is its contrapositive?

• If  $k+1$ is irrational, then $k$ is also irrational. 

• If $k$ is rational, then  $k+1$   is irrational.

• If $k+1$ is irrational, then  $k$  is rational. 

• If $k+1$ is irrational, then  $k$  is irrational.

Complete the argument below by supplying the missing statement.
Given: If two integers are both odd, their product is also odd.

Proof: Assume that  $x$ and  $y$  are both odd integers.

Let  $x=2m+1$  and  $y=2n+1$  for some integers  $m\ and\ n$ .
 $xy=\left(2m+1\right)\left(2n+1\right)$  
 $xy=4mn+2m+2n+1$  
 $xy=$  _____________________

•  $2mn+1$  

•  $2\left(2mn+m+n+1\right)$  

•  $2\left(2mn+1\right)$  

•  $2mn$  

Complete the argument below by supplying the missing statement.
Given: Show that the sum of two odd integers is an even integer.


Proof:
An odd integer is represented by  $2k+1$ .
Let  $2m+1\ and\ 2n+1$  be the odd integers.
 $\left(2m+1\right)+\left(2n+1\right)=2m+2n+2$  
 $\left(2m+1\right)+\left(2n+1\right)=2\left(m+n+1\right)$  
Let  $m+n+1$  equal to $k$ .
_____________________ = _______________________

•  $\left(2m+1\right)+\left(2n+1\right)=2k$  

•  $\left(2m+1\right)+\left(2n+1\right)=2k+1$  

•  $\left(2m+1\right)+\left(2n+1\right)=2k+m+1$  

•  $\left(2m+1\right)+\left(2n+1\right)=2k+m+n+1$  

Tags

CCSS.6.EE.B.6

 What conclusion can you make from the given proof below?
Given: Show that  $\log\ 2$  is an irrational number.

Proof: Suppose  $\log2$ is a rational number. Let  $\log2=\frac{a}{b}$ where  $GCF(a,b)=1.$ Then, 
 $2=10^{\frac{a}{b}}$ 
 $2^b=10^a$  
 $2^b=2^a\cdot5^a$  
It follows that  $2^{\left(b-a\right)}=5^a$ .

• This is impossible to happen. Thus, the assumption is false and so log 2 is an irrational number.

• This is impossible to happen. Thus, the assumption is false and so log 2 is a rational number.c

• This is possible to happen. Thus, the assumption is true and so log 2 is an irrational number.

• This is possible to happen. Thus, the assumption is true and so log 2 is a rational number.

Tags

CCSS.HSN.RN.B.3