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alexs1000
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Homework Statement
Prove that(power set) P(E) U P(F) is a subset of P(E U F)
Homework Equations
P(E) U P(F) is a subset of P(E U F)
The Attempt at a Solution
P(E)U P(F)={x:xεP(E) or xεP(F)}
but P(E)={X:X is a subset of E} or P(E)={x:xεX→xεE}
so we get P(E)U P(F)={x:xεX→xεE or xεX→xεF}
but logical implication p→q⇔non(p) or q
so we get P(E) U P(F)={x:non(xεX) or xεE or non(xεX) or xεF}
therefore by the idempotence property and comutativity of the logical operators
P(E) U P(F)={x:non(xεX) or xεE or xεF}
and we get P(E) U P(F)={x:non(xεX) or xεE U F}
which is exactly P(E U F).My question is why is it that only the inclusion stands, and why not equality also.Thank you.