THX Theorem. Suppose A, B and C are sets, A and B are not disjoint, A and C are

Question

THX

Theorem. Suppose A, B and C are sets, A and B are not disjoint, A and C are not disjoint, and A has exactly one element. Then B and C are not disjoint. Proof. Since A and B are not disjoint, we can let b be something such that be and Similarly, since A and C are not disjoint, there is some object c such that and Since A has only one element, we must have b = c. Thus and therefore B and C are not disjoint. Prove that for every real number x there is a unique real number y such that x2y = x - y. Prove that there is a unique real number x such that for every real number y, xy + x - 4 = 4y. Prove that for every real number x, if and then there is a unique real number y such that y/x = y - x. Prove that for every real number x, if then there is a unique real number y such that for every real number z, zy = z/x. Recall that if F is a family of sets, then Suppose we define a new set U!F by the formula

Explanation / Answer

2)

xy+x-4=4y

xy+x = 4+4y

x(1+y) = 4+4y

x = (4+4y)/(1+y)

For a single value of y, there exists only one value of x

Hence proved

3)

y/x = y-x

y = xy - x^2

xy-y = x^2

y(x-1) = x^2

y = x^2 / (x-1)

For a single value of x, there exists only one value of y

Hence proved

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