Pls help explain the solution of question 7

[blazing slate]

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[silver brook]

Pls help explain the solution of question 9, 11, 13, 15, 17 and 19, instead of 7

[blissful slate]

for question 9, do you understand what the cartesian product of two sets is? @silver brook

[silver brook]

Actually I don’t have a clear understanding of Cartesian product

[blissful slate]

the set you get from taking the cartesian product of two sets contains all ordered pairs of elements from each individual set

so if set A has a as an element and set B has b, then A x B will contain (a,b) as an element

do you think you can work out what the set {a,b} x {0} is from that?

[silver brook]

Thank you for your explanation. I think I don’t understand why there’s a ( ). Cuz I have been using { } a lot to represent a set. I don’t know what the ( ) means…

[blissful slate]

in the context of the cartesian product, the () brackets denote an ordered pair, which is a 2-tuple

tuples are just lists of mathematical objects, and generally () brackets are used to denote them

so when you take the cartesian product, you get 2-tuples with the first element from the first set and the second from the second, hence the round brackets

[silver brook]

I guess I understand question 9 now…thank you…and how to solve question 11?

[blissful slate]

the first step is computing the power set of {1,2,3}, are you ok with doing this?

[silver brook]

Yes, I am ok with this

[blissful slate]

the next step is understanding what the set in the question is, can you explain what elements are in the set?

(sorry for late reply, fell asleep lol)

[silver brook]

I don’t understand your question🥹
(It’s alrighttttt)

[blissful slate]

what actually is the set in question 11? what are its elements?

[silver brook]

This the set but I don’t know how to do “such that the number of elements in X is less than or equal to 1”

[blissful slate]

the set in the question is asking for subsets of this set with <=1 elements, can you think of any?

[silver brook]

These four

[blissful slate]

they do all have cardinality <=1, but as youve seen in the solution is that there are a few more

as an example, whats the cardinality of the set {a,b,c}?

[silver brook]

The cardinality of the set {a,b,c} is 3

The cardinality of the set {1,2} is 2
The cardinality of the set {1,3} is 2
The cardinality of the set {2,3} is 2
The cardinality of the set {1,2,3} is 3

[blissful slate]

and what about the set {{a,b,c}}?

[blissful slate]

or the set {{a,b,c}, {d,e,f}}?

[silver brook]

The cardinality is 1

[silver brook]

The cardinality is 2

[blissful slate]

yeah exactly, so can you find any more cardinality 1 subsets of the powerset above?

[silver brook]

These?

[blissful slate]

yep, thats them

so thats how you get to the solution of q11

[blissful slate]

the empty set is a subset of the powerset (bc its a subset of every set), and it has cardinality 0

[silver brook]

Got it. Thanks!🥹

[ivory juniper]

@silver brook

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