1. (50 pt., 10 pt. each) Prove each of the following statements using a direct p
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1. (50 pt., 10 pt. each) 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 x. The difference of any rational number and any irrational number is irrational. For all real numbers x and y, min(x,y) = If you pick four socks from a drawer containing just white socks, blue socks and black c. d. xty-ryl and max(x, y) = x+y2 e. socks, you must get at least one pair of white socks, blue socks or black socks.Explanation / Answer
a) Direct proof - first of all, let me tell you parity is the property, on which you can say any integer is even or odd/
suppose n is an integer then (n-1) and (n+1) will be its consicutive integer. and if we say n is even and n=2*a(where a is some random integer),so (n+1)=(2*a)+1 and (n-1)=(2*a)-1 will be odd, which both are consicutive to integer 'n'. And same is true with opposite case, if n is odd then (n+1) and (n-1) will be even.
b) Direct proof - As we know about transitive propoerty, which says if a=b and b=c then a=c in which we user '=' operator as an action but this property will hold true for any operator because of transitive behaviour, so if we change '=' operator with division operator then also it will be true like if y/x and z/y then z/x.
c) Contradiction suppose x is rational number and y is irrational number and by contradiction we assume that x - y is rational, and as we know summation of 2 rational number is also a rational number and if we perform x - y +(-x) so this will be rational here -x is also rational . x-y+(-x) gives -y as rational(so y will also be rational) but we already assumed y as irrational, which became false statement so we can say that the difference of any rational and irrational number is irrational.
d) Direct proof - for all real numbers x,y; min(x,y) = ((x+y)-(|x-y|))/2 and max(x,y) = ((x+y)+(|x-y|))/2.
case for min(x,y) --- suppose x is one number and other number is x-1,which is clearly less than x. so
min(x,x-1) = ((x+(x-1))-(|x-(x-1)|))/2
= ((2x-1)-1)/2
= (2x-2)/2 i.e. x-1 so it is proved for min(x,y)
now for max(x,y); x is one number and other number is x+1 which is larger than x
max(x,(x+1)) = ((x+(x+1))+(|x-(x+1)|))/2
=((2x+1)+1)/2 = x+1 so it is proved for both the cases that
for all real numbers x,y; min(x,y) = ((x+y)-(|x-y|))/2 and max(x,y) = ((x+y)+(|x-y|))/2.
e) direct proof if we pick 4 socks from a drawer containg just white,blue and black socks one by one then in 100 % fault rate also, we will get 3 different color sock and one sock left to pick and as we know pair is a group of two so if we get any of the sock with any probability will get atleast one pair of socks with us, it was the worst case.
hope, u got your solution and feel free to ask any qyery and to give feedback.
Thank you
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