#discrete maths

1 messages · Page 1 of 1 (latest)

wraith rock
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in the x,y in the set containing 15, would both x and y be equal to 15?

unknown.png
wintry helm
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possibilities are
x = 15
y = ø
or
x = ø
y = 15

wraith rock
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So then Z|y|z

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Would be z|1??

summer bronze
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isn't that the set of pairs (x, y) in {15} x Z such that y | x ?

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otherwise i don't understand that notation, seems like there's a subscript typo there

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if so, then ofc (x, y) is a pair in the set iff x=15 and y is an integer divisor of 15 (finitely many possibilities that you should know how to list)

hybrid slate
hybrid slate
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I think I'd have used a colon in place of the first line, though.

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If I'm reading it right, then it's the set of all ordered pairs (x, y) in the Cartesian product of {15} and Z such that y divides x.

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Now, the Cartesian product A x B, as I mentioned before to someone else, is the set of all ordered pairs (x, y) such that x is an element of A and y is an element of B.

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So the Cartesian product {15} x Z would be the set of all ordered pairs (x, y) such that x is in {15} and y is in Z.

hybrid slate
wraith rock
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so any number that divides y|x but im confused what y|x is since theres only 2 things in the set

summer bronze
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that Z is the set of integers, the first bar "|" means "such that", not divisibility, the second one is divisibility

wraith rock
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ahhh i see

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but would the ints only include only 156?

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15*

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cause x,y is only in the set containing 15

summer bronze
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no, Z is the set of all integers

summer bronze
wraith rock
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then what would y be?

summer bronze
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for each pair (x, y) in that set, (x, y) is in {15} x Z, meaning x is in {15} and y is in Z, so x=15; but remember that they also satisfy y | x, aka y | 15

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so y is any integer divisor of 15

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do you know what those are? there are finitely many possibilities

wraith rock
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uhhh

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1 15 3 5

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so then cardinality would be 4?

summer bronze
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what about -1?

wraith rock
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oh

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then 8

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i forgot about those

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hahaha

summer bronze
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yep

wraith rock
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thanks

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what about the cardinality of a number range in R

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so like [1,5]

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that would probably be N0

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since there are infinite amoiunts

summer bronze
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if you mean the closed interval [1, 5] of real numbers, then no, they're c = 2^aleph_0 in cardinality

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which is strictly more than aleph_0

wraith rock
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would it be beccause u get the points from R^n

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when n = 2?

summer bronze
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no that is just because it's the cardinality of the set of subsets of the natural numbers

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and that is the same as the cardinal 2^{|N|}

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for example, the cardinality of the set of subsets of {1, 2, ..., n}
is 2^n

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that's because you can do a counting, for each element from {1, 2, ..., n} you decide if you're gonna put in the subset or not (two choices), and in the end you get your subset uniquely determined by those choices

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so that's two choices for each of the n elements, hence 2^n subsets in total

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you can apply this reasoning to prove the same for the set of subsets of N

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and there is a definition of power of cardinals

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given k and m two cardinals, there is a definition for the cardinal k^m (it's actually just the cardinality of a set K^M where K and M are sets of cardinality k and m, respectively)

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and then 2^aleph_0 is well defined and equal to the cardinality of the set of subsets of N, and this latter one can be shown to be the cardinality of the set of real numbers (you use binary representation)

wraith rock
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so then would that look like

unknown.png
chilly magnet
hybrid slate
summer bronze
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I'm using latex syntax

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so what i meant is what you typed

summer bronze
wraith rock
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alr thanks

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:))

wintry helm
wraith rock
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i thnk i figured it out

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its basically asking for factors of 15

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so its saying x,y are elements of the set {15}

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but then its using all numbers in the ints

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so its y = Z and x=15

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which is z|15

hybrid slate
# wraith rock so its saying x,y are elements of the set {15}

No, x and y are not elements of the set {15}. I've already explained this a couple of times. (x, y) is an element of the Cartesian product {15} x Z, meaning it's an element of the set of all ordered pairs of which the first element of the ordered pair is an element of {15} and the second element of the ordered pair is an element of Z.