I am not quite sure but is the following statement true or false:
"two planes (twodimensional) cannot intersect in a point"
I would say the statement is false because if two planes intersect then they intersect in a line which consists of infinitely many points (like the intersection of the xy- and yz-plane is the y-axis).
But the solution says the statement is true and I don't understand why. What could be the reason?
On
Oh, I see what you thought the statement was saying!
"two lines cannot intersect IN a point"
The "IN" means that the entire intersection is the single point.
So this sentence is saying: It's not possible for two planes to intersect at one and only one point.
It is NOT saying: Two planes can never have any points of intersection.
I think you interpretted it as the latter-- not the former.
Of course planes intersect at points. Everything consists of points so if things intersect they must intersect at points. What else could they intersect at? Rabbits?
But HOW many points? If they are "flat" planes it's either infinite or not at all. It can't be just one.
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You think you believe the statement is false.
But you are mistaken; you actually believe the statment is true.
Reread the statement.
The statement is "two planes (twodimensional) canNOT intersect in a point"
You say "if two planes intersect then they intersect in a line which consists of infinitely many points"
That's an argument for why the statement is TRUE; not why it is false. If it were false the planes COULD intersect at a point. You believe they can't, which is exactly what the statement says.
So you actually believe the statement is TRUE.
Actually, as stated the question is ambiguous. Yes, in a $3$-dimensional spaces two distinct planes either don't intersect or their intersection is a line. But in dimension $4$ or higher, their intersection may well consist of a single point.