You must write down all proofs in acceptable mathematical language: make sure yo

Question

You must write down all proofs in acceptable mathematical language: make sure you mark the beginning and end of the proof, define all variables, use complete, grammatically correct sentences, and give a justification for each assertion (e.g., by definition of...). See lecture slides for examples. Definitions: An integer n is even if and only if there exists an integer k such that n 2k An integer n is odd if and only if there exists an integer k such that n = 2k + 1. they are both odd Two integers have opposite parity when one is even and the other one is odd 0, denoted d | n, if and only if there exists an An integer n is divisible by an integer d with d integer k such that n = dk. 0 such that A real number r is rational if and only if there exist integers a and b with b r = a/b. For any real number x, the absolute value of x, denoted , is defined as follows: . x ifx 20 la' = {.iif

Explanation / Answer

(e)If xy then max(x,y)=x and
(x+y+|xy|)/2=(x+y+xy)/2
=> 2x/2=x.
Else, then max(x,y)=y and
(x+y+|xy|)/2=(x+y+yx)/2
=> =2y/2
=> =y.
Thus, max(x,y)=(x+y+|xy|)/2

Now min proof

If xy then min(x,y)=y and
(x+y-|xy|)/2=(x+y-x+y)/2
=> 2y/2=y.
Else, then min(x,y)=x and
(x+y-|xy|)/2=(x+y-y+x)/2
=> =2x/2
=> =x.
Thus, min(x,y)=(x+y-|xy|)/2

(f)  We prove first the direct implication.
Assume n is even.
Then n = 2k for some integer k.
Then 7n + 4 = 14k + 4 = 2(7k + 2), which is even.
For the converse, which is “if 7n + 4 is even, then n is even”, we use a proof by contrapositive.
The contrapositive is: “if n is not even
(that is, odd),
then 7n + 4 is not even (that is, odd)”.
If n is odd, then n = 2k + 1, for some integer k.
Then 7n + 4 = 14k + 11 = 2(7k + 5) + 1,
which is odd.

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