Prove the following lemma:
Let A,B.C. and D be sets. If A( B and D ( C, then A/C ( B/D
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)
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