Math Is Fun Forum

Median Joe

Member

Registered: 2019-12-04

Posts: 8

Set Theory problem

Given that A = {1,2,3,4}, calculate the elements of the sets B andC, given the following facts:

1) A = BC
2) A'C = {5,6}
3) (A + B)C'  = {7}

I know I could have used LaTeX for the set notation but I think the boolean algebra notation is easier to read.
So juxtaposition is set intersection, "+" is set union, and a primed symbol (') is set complement.

I think I've solved it but the solution seems a bit long-winded, so I would like to see other solutions which may be simpler.

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“I like work: it fascinates me. I can sit and look at it for hours.”

Bob

Administrator

Registered: 2010-06-20

Posts: 10,145

Re: Set Theory problem

hi Median Joe,

I thought I'd try a Venn diagram.

Draw a circle to represent B and another overlapping it to represent C.

Put 1,2,3 and 4 into the overlap as we know A = BC.

A'C is the part of C that is not in A, so you can put in 5 and 6.

Finally A + B is just B, as all A is in B, so the third fact reduces to BC' = {7} ie the parts of B that are not in C so you can put in 7.

Then all you need to do is look at what is in B and what is in C.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Median Joe

Member

Registered: 2019-12-04

Posts: 8

Re: Set Theory problem

Hi Bob,

Thanks for the reply. I hadn't thought of doing a Venn diagram, but that's probably the easiest and most intuitive solution.

For the record, here is my algebraic solution.

1) A = BC
2) A'C = {5,6}
3) (A + B)C'  = {7}

Taking the complement of both sides of (1) gives A' = (BC)' = B'+C' (by de Morgans' rule)
Substituting this result into (2) gives (B'+C')C = B'C = {5,6}.

Substituting (1) into (3) gives (BC + B)C' = {7}, but BC + B = B, therefore BC' = {7}.

Now, B = BC + BC'
So B = A + {7} = {1,2,3,4} + {7} = {1,2,3,4,7}

and C = BC + B'C
So C = A + {5,6} = {1,2,3,4} + {5,6} = {1,2,3,4,5,6}

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“I like work: it fascinates me. I can sit and look at it for hours.”