Prove the following lemma

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Prove the following lemma:

Let A,B.C. and D be sets. If A( B and D ( C, then A/C ( B/D

Feb 6th, 2015

Please note that we have the subsequent logical implications:

1) if x belongs to A, then x belongs to A union B

now I consider an element which belongs to A/C, namely:

x belongs to A and x doesn't belong to C, the we can write using 1):

(x is in A or x is in B) and x is not in C, which is equivalent to:

x is in A and x is not in C (this statement is known)

2) or x is in B and x is not in C ( this is a new statement)

Now please note that:

since D is a sub set of C, then we have:

if x is in D then x is in C,              which is equivalent to this one:

3) if x is not in C then x is not in D

Next I re-write statement 2) using the statement 3), and I get:

x is in B and x is not in C   then   x is in B and x is not in D,

so x belongs to B/D

Reassuming I proved that:

if x belongs to A/C then x belongs to B/D, or:

(A/C) is a subset of (B/D)

Feb 6th, 2015

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