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hrcatena
11.03.2020 •
Mathematics
Prove the following statement by contrapositive or by contradiction:
For all integers m and n, if m + n is even then m, n are both even or m, n are both odd.
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Ответ:
Proof by Contrapositive
The contraposition of the statement is
If 'm is odd and n is even' or 'm is even and n is odd' then 'm+n is odd'.
a)At first we will take one of the statement as 'm is odd and n is even'
As we know the ways to write a even and odd number
m is odd mean
m=2a+1,
n=2b,
where a and b are any integers
Now substitute the values of m and n in (m+n)
m+n = (2a+1)+2b = 2a+2b+1 = 2(a+b)+1 Suppose c =a+b
m+n = 2c+1, which is odd
Since the contrapostive statement is true than the original statement is also true.
Similarly we can take m is even and n is odd, for that
m=2a and n=2b+1, substitute the values of m and n in m+n, we get
m+n = 2a+(2b+1) = 2a+2b+1 = 2(a+b)+1 Suppose c =a+b
m+n = 2c+1, which is odd
Again, Since the contrapostive statement is true than the original statement is also true.
Ответ:
We can represent the months since haircut along x-axis and the inches of grown along y-axis.
Please refer to the attached table.