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Adv Alegbra HW Solutions 40

# Adv Alegbra HW Solutions 40 - 40 2.5 Let A and B be sets...

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40 2.5 Let A and B be sets, and let a A and b B . De fi ne their ordered pair as follows: ( a , b ) = { a , { a , b }} . If a A and b B , prove that ( a , b ) = ( a , b ) if and only if a = a and b = b . Solution. The result is obviously true if a = a and b = b . For the converse, assume that { a { a , b }} = { a { a , b }} There are two cases: a = a and { a , b } = { a , b }; a = { a , b } and { a , b } = a . If a = a , we have { a , b } = { a , b } = { a , b } . Therefore, { a , b } − { a } = { a , b } − { a } . If a = b , the left side is empty, hence the right side is also empty, and so a = b ; therefore, b = b . If a = b , the the left side is { b } , and so the right side is nonempty and is equal to { b } . Therefore, b = b , as desired. In the second case, a = { a , b } = {{ a , b } b } . Hence, a ∈ { a , b } and { a , b } ∈ {{ a , b } , b } = a , contradicting the axiom a x a being false. Therefore, this case cannot occur. 2.6 Let = { ( x , x ) : x R } ; thus, is the line in the plane which passes
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