Which of the following statements is always true ? Select one: O a. If A, B, and C are sets with AC BCC, and A and B are countable, then C is countable. O b. There exists an injective function f: R→ Q. O c. Let S = {(a, b) ≤ N× N: a ≤ b²}. Then S is countably infinite. Every infinite subset of R is countable. O d. e. If A and B are two sets such that A is countable and A< B, then B is uncountable.

Q: D. Let A, B, and C be subsets of some universal set U (a) If ANBC C and A° N B C C, prove that B CC

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Q: e) Does the equality |A U B| = |A| + |B| hold for every set A and B? f) Is it true that for any…

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Q: F and H are sets of real numbers defined as follows. F={w|w<3} H={w | w≤9} Write FU H and FN H using…

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Q: If A and B are nonempty, bounded, and satisfy A c B then sup A < sup B. If sup A < inf B for sets A…

A: a) Given that A,B are non empty sets, and A⊂B The objective is to show that supA≤sup B Since it is…

Q: (a) Let [a, b], a <b, be any closed interval. Prove that |[a, b]| = |[0, 1].

A: Let  be any closed interval.The main aim is to prove that :"As per the policy we are allowed to…

Q: Let V; = R for each positive integer i. Find each of the following. (Enter your answers using…

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Q: Let a, b ∈ R. Show if a ≤ b1 for every b1 > b, then a ≤ b.

A: Density Property of ℝ If x,y are real numbers with x&lt;y, then ∃ a rational number r such that…

Q: Proficiency #1. [Statements] Which of the following sentences are statements? Do not worry about…

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Q: 48

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Q: Suppose F and G are families of sets. Prove that (UF)n (UG) ≤ U(F N G) iff VA EFVBE G(An BCU (FNG)).

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Q: 2.4. Let A and B be sets of real numbers, let f be a function from R to R, and let P be the set of…

A: As per the policy, we are solving three parts. Please repost it and specify which part is to be…

Q: Let R = {x € Z | x is divisible by 2}. Determine which of the following sets are subsets of R.…

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Q: Find the cardinality of each of the following sets. a. A = {x:xe Z and x² = -9} b. B = {x:x € R and…

A: The given sets,(a) (b) (c) (d) The aim is to find the cardinality of each of the given sets.

Q: Let P denote the set of integers strictly greater than 1. For i> 2, define X;= {ik|k e P}. Describe…

A: Given P=n|n∈ℕ,n&gt;1 Then,  Xi=ik|k∈P=multiples of i Claim: ∪i=2∞Xi=Set of Composite numebrs Proof:…

Q: Which of the following sets are equal? A = {b, g, j} B= {x, y} C= j, g, b} D= {x|1sx<3 and xEN} J=…

A: Here, A = {b, g, j} B = {x, y} C = {b, g, j} D = {1, 2, 3} J = {0, 2, 4, 6, .....} K = {6, 8, 10,…