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Answer true or false for each statement and give a logically valid explanation to support your...

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Advance Math

Answer true or false for each statement and give a logicallyvalid explanation to support your answer (you can give acounterexample if it is false, or make a proof if it is true).

a. The multiplication of two irrational numbers is an irrationalnumber.

b. The set {( 1?n/ n , 8n+1/ n ) |n ? N } is an open covering of(?1, 8].

c. The sequence {( 1?3n /n + (?1)n ) |n ? N } has asub-sequence that converges to -4.

d. The sum of the Lebesgue measure of the sets A = C ? [?5, 1),and B = Q ? [4, 10] is 6, where C is the Cantor ternary set, and Qis the rational numbers set.

e. Find the inf and sup of the set A = {x ? R : x 2 ?4x ? 20 ? 5x} are -5 and 4 respectively.

f. The intersection set for the following collection ofintervals is open. {(?1/n ? 2, 3 + 1/n), n ? N} .

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