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a logic question

Started by TimChan Oct 19, 2005 at 12:57 PM 10 replies 900+ views

Original Post

Oct 19, 2005 at 12:57 PM


We say that person A dominates person B if and only if A is taller than B and A is heavier than B.
Which of the following statements MUST be true?

(i)	If A dominates B and B dominates C, then A dominates C.
(ii)	If A dominates B, then B must not dominate A.
(iii)	If A does not dominate B, then B must dominate A.

A.	(ii) only
B.	(i) and (iii) only
C.	(ii) and (iii) only
D.	None of the above

Statement(iii) is of coz incorrect so the option B is incorrect. But why is statement(i) not true?

Nahoopii

Oct 19, 2005 at 1:00 PM

Isnt (i) true?
If A is heavier and taller than B, B is heavier and taller than C, isn't A heavier and taller than C?

Dmytry

264

Dmytry

Oct 19, 2005 at 1:03 PM

answer: none of the above.
i and ii must be true.

Proof for i: replacing the "dominate" by meaning of dominate, we get:
(A is heavier than B AND
A is taller than B) AND
(B is heavier than C AND
B is taller than C)
=
A is heavier than B AND
B is heavier than C AND
A is taller than B AND
B is taller than C

From first two it follows that A is heavier than C. From second two it follows that A is taller than C. Thus A dominates C.

iii is not necessarily true because a can be heavier AND shorter than b . In this case neither dominates.

kryat

Oct 19, 2005 at 1:04 PM

statement i is true...

b is incorrect because true and false = false.

TimChan

Author

TimChan

Oct 19, 2005 at 1:05 PM

Quote:
Original post by Rixter
Does it say anything about if they are the same height and weight?

if they are same height and weight, they do not dominate each other.

Rixter

151

Rixter

Oct 19, 2005 at 1:06 PM

Quote:
Original post by TimChan
Quote:
Original post by Rixter
Does it say anything about if they are the same height and weight?

if they are same height and weight, they do not dominate each other.

Right, I got mixed up on which was number iii :)

Thermodynamics

Oct 19, 2005 at 1:08 PM

The first two replys are true. Therefore ... I got nuttin to add.

TimChan

Author

TimChan

Oct 19, 2005 at 1:09 PM

oops, the answer is D...

ApochPiQ

1,378

ApochPiQ

Oct 19, 2005 at 1:10 PM

Dominate(A,B) = (Height(A) > Height(B) && Weight(A) > Weight(B))

i. If Dominate(A,B) && Dominate(B,C) Then Dominate(A,C)
or: If (Height(A) > Height(B) && Weight(A) > Weight(B)) && (Height(B) > Height(C) && Weight(B) > Weight(C)) Then (Height(A) > Height(C) && Weight(A) > Weight(C))
This is the transitive property of inequality in a slightly more complex form. This is a valid proposition.

ii. If Dominate(A, B) then ~Dominate(B, A)
or: If (Height(A) > Height(B) && Weight(A) > Weight(B)) Then ~((Height(B) > Height(A) && Weight(B) > Weight(A))
or: If (Height(A) > Height(B) && Weight(A) > Weight(B)) Then (Height(B) < Height(A) || Weight(B) < Weight(A))
Since the conditional part of the proposition asserts that Height(A) > Height(B), it is necessary that Height(B) < Height(A). Therefore, the resultant part of the proposition is true by default. This is a valid proposition by virtue of tautology.

iii. If ~Dominate(A, B) then Dominate(B, A)
or: If ~(Height(A) > Height(B) && Weight(A) > Weight(B)) Then (Height(B) > Height(A) && Weight(B) > Weight(A))
or: If (Height(A) <= Height(B) || Weight(A) <= Weight(B)) Then (Height(B) > Height(A) && Weight(B) > Weight(A))
Counterexample: Height(A) = Height(B)
Counterexample: Weight(A) = Weight(B)
This is not a valid proposition.

(i) and (ii) are valid, (iii) is not. Answer is D.