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<a https://www.thestudentroom.co.uk/showthread.php?t=7467335'> Year 12s: what will be the first way you research your future options? >>
<a https://www.thestudentroom.co.uk/showthread.php?t=7467335'> Year 12s: what will be the first way you research your future options? >>

set theory help

Hi

I'm starting uni soon and I thought I would study a bit in advance so got Geoff smiths algebra and analysis book, and I'm stuck on the following questions (to be fair it does say they're harder as a warning...)

At the back it says (a) is n x 2^(n-1)... I understand n is the cardinality of Set a (n elements) but I don't know how the cardinality of S is 2^(n-1)... I thought it would be 2^n
(edited 9 years ago)
DSC_0107.jpg509.3KB
Well, ∅∈P(A)\varnothing \in P(A) so I guess that rules out some of the possibilities.
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hi1345
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Original post by atsruser
Well, ∅∈P(A)\varnothing \in P(A) so I guess that rules out some of the possibilities.

At the back it says the cardinality of S \ {a} is 2^(n-1)... Does that mean the power set of a minus the individual elements in a? Sorry I'm a little confused and Wikipedia doesn't help much :P
Original post by hi1345
At the back it says the cardinality of S \ {a} is 2^(n-1)... Does that mean the power set of a minus the individual elements in a? Sorry I'm a little confused and Wikipedia doesn't help much :P


If S is a set of n elements, and "a" is one of those elements then S \ {a} is the set containing all the elements of S excluding a. There are n-1 elements in S \ {a}, and 2^(n-1) elements in the powerset of S \ {a}.

"a" is an element
{a} is a set of just one element - "a".
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hi1345
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Original post by ghostwalker
If S is a set of n elements, and "a" is one of those elements then S \ {a} is the set containing all the elements of S excluding a. There are n-1 elements in S \ {a}, and 2^(n-1) elements in the powerset of S \ {a}.

"a" is an element
{a} is a set of just one element - "a".


Thanks I understand pt (a) and (b) now although have no idea about (c) or (d)
Original post by hi1345
Thanks I understand pt (a) and (b) now although have no idea about (c) or (d)


Rather a big hint for c) so I'll put it in a spoiler.

Spoiler



That should give you a clue as to how to do part d) too.
(edited 9 years ago)
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hi1345
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Original post by ghostwalker
Rather a big hint for c) so I'll put it in a spoiler.

Spoiler



That should give you a clue as to how to do part d) too.


Thanks for the clear explanation, this stuff is a lot harder than A level 😂

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