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Mohamed G. Gouda						CS 311 
Summer 2014							Homework 3
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1. (4 points) 
Use direct inference to prove the predicate 
	(A sub AvB)
where A and B are sets, "sub" denotes "subset", and "v" denotes the set union 
operator. 

(Hint: The first predicate in the proof is (x in A) and the last predicate 
in the proof is (x in AvB).)
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2. (2 points)
Use direct inference to prove the predicate 
	(AvB sub A) => (B sub A)
where A and B are sets, "sub" denotes "subset", and "v" denotes the set union 
operator. 

(Hint: The first predicate in the proof is (x in B) and the last predicate 
in the proof is (x in A).)
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3. (4 points)
Use direct inference to prove the predicate 
	(AvB = A) => (B sub A)
where A and B are sets, "sub" denotes "subset", and "v" denotes the set union 
operator. 

(Hint: The first predicate in the proof is (AvB = A) and the last predicate 
in the proof is (B sub A).)
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Solutions:
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1.
	(x in A)
=> {P => (P or F)}
	(x in A) or ((x in B) and not(x in B))
=> {distribution of and over or}
	((x in A) or (x in B)) and
	((x in A) or not(x in B))
=> {(P and Q) => P}
	((x in A) or (x in B))
=> {definition of the v set operattor}
	(x in AvB)
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2.
	(x in B)
=> {(AvB sub A) is assumed to be T}
	(x in B) and (AvB sub A)
=> {definition of (AvB sub A)}
	(x in B) and
	(All y, (y in AvB) => (y in A))
=> {choosing y to be x}
	(x in B) and
	((x in AvB) => (x in A))
=> {definition of AvB}
	(x in B) and
	((x in A or x in B) => (x in A))
=> {from (P or Q)=> R infer ((P=>R) and (Q=>R))}
	(x in B) and
	((x in A) => (x in A)) and
	((x in B) => (x in A)) 
=> {from P and (P => Q) infer Q}
	(x in A)
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3.
	(AvB = A)
=> {definition of =}
	(AvB sub A) and (A sub AvB)
=> {Problem 2: (AvB sub A) => (B sub A)}
	(B sub A) and (A sub AvB)
=> {(P and Q) => P}
	(B sub A)
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