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3 years ago

Let A and B be bounded subsets of R. (a) Why does sup(AUB) exist? (b) Prove that sup(AUB) = max

Mathematics

1 answer:

makvit [3.9K]3 years ago

3 0

Answer:

a) sup(AUB) exist because A and B are bounded.

The definition of sup(A)={x∈A/y∈A,y≤x}

If x=sup(A),x∈A ⇒ x∈(A∪B)

If z=sup(B), z∈B ⇒ z∈(A∪B)

b)The value of sup(A∪B)=max(sup(B),sup(A))

proof

x∈A∪B⇒x∈A ∨ x∈B⇒x≤sup(A) ∨ x≤sup(B) then x≤max{sup(A),sup(B)}

sup(A)≤sup(A∪B) and

sup(B)≤sup(A∪B)

By definition of max:

max{sup(A),sup(B)}≤sup(A∪B).

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Answer:

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Solution:

The vertex form of quadratic equation is generally given as,

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From question, given that $x^{2}-6 x+8=0$ .

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Let us first convert the given quadratic equation to vertex form (eqn 1)

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By comparing $(x-3)^{2}-1=0$ with $a(x-h)^{2}+k$

we get the values of (h,k)

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