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Topic: set builder notation
Replies: 12   Last Post: Aug 24, 2013 1:38 AM

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Graham Cooper

Posts: 4,495
Registered: 5/20/10
Re: set builder notation
Posted: Aug 17, 2013 7:34 AM
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On Saturday, August 17, 2013 4:12:34 AM UTC-7, wrote:
> S = {x /in A | P(x) }
> For the set builder notation above, what we really means is:
> all things x, such that "x is element of A *and* P(x) is true" correct?
> The vertical bar is essentially conjunction, correct?/

No, consider

S = { x /in A | P(a,b,x,y,z) }

there is an implicit ALL(x)
by placing x on the left of the bar.

x is no longer free in P

the conjunction A&B
just returns a value of true or false.

S = {x | p(x)}

is shorthand for

ALL(x) xeS <-> p(x)

Putting complex expressions before the bar is a different shorthand...

it's really

S = { x | xeA & P(x) }

Set Specification increases the level of the logic above FOL.

Now you can solve S UNION T

using the pre-designated parameter in p(...)

X e ( S U T ) <- X e S
X e ( S U T ) <- X e T

so to test for UNION it will then backward chain to
a direct element check on one or both of the unioned sets

...via how they were specified... and then call p()

X e S <- p(X)



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