1)Prove each of the following statements using a direct proof, a proof by contra
ID: 3872037 • Letter: 1
Question
1)Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. Indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusion (what you must show) of the proof. For direct proofs, indicate the statement to be proven in the form “if… then.” For proofs by contrapositive, indicate the contrapositive of the statement to be proven in the form “if… then.” For proofs by contradiction, indicate the negation of the statement to be proven. For proofs by cases, indicate all possible cases.
a. Any two consecutive integers have opposite parity.
b. For all integers x, y, and z, if y is divisible by x and z is divisible by y, then z is divisible by x.
c. The difference of any rational number and any irrational number is irrational.
d. For all real numbers x and y, min(x,y) = [x+y|x-y|]/2 and max(x,y) = [x+y+|x-y|]/2 .
e. If you pick four socks from a drawer containing just white socks, blue socks and black socks, you must get at least one pair of white socks, blue socks or black socks.
Explanation / Answer
a. Any two consecutive numbers differ by +1 or -1. Whenever we add or subtract this we will get a number that is of opposite partity of that number. A number will have a parity of even or odd numbers of 1, so odd +1 = even, and even + 1 = odd. It doesn't matter if there is carry or borrow but this always holds.
b. If y is divisible x that means y=kx, where k is any integer. Similary z=ly where l is any integer. So z=lkx where lk can be written as p so z=px meaning z is divisible by x.
c. Suppose r is irrational and x is rational
To prove r+x is irrational
Proof by contradiction: Lets say r+x is rational
Since x is rational then -r must be rational.
So r+x -r would be rational by our hypthesis
But r+x-r=x which is irrational but our hypothesis says it is rational hence proved that our hypothesis is wrong therefore r+x is irrational.
d. Lets say x is smaller and y is greater.
So (x+y-(x-y))/2
Lets say ((x-abs(x-y))+y)/2 This is absolutely valid and since x is greater and y is smaller so the result would be x-y
(x-x+y+y)/2
=2y/2
=y.
=min(x,y)
(x+y+abs(x-y))/2
Again abs(x-y) would lead x-y
Then (x+y+x-y)/2
2x/2
=x
=max(x,y)
This works the other way around as well where abs(x-y) gives y-x.
e. If we have picked 3 socks there can be two cases either we already have a pair, so we are finished or we have 3 socks all of different colors. So the next sock would either be white, blue or black and picking any one of them results us in having a pair of which ever color we have.
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