3. Every nonzero rational number can be written as the product of three irrational numbers. For nonempty sets A, B, and C, let f: A -> B and g : B -> C. If g is onto, then gof is onto. 4.

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Q: 2. Show that the limit set of the set of rational numbers, i.e. Q', is the set of real numbers R,…

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Q: 30. If S is the set of negative integers sup S does not exists sup S = 0 sup S is any nonnegative…

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Q: 26. Let I be the set of all infinite sets. Is I an element of itself?

A: Given I is the set of all infinite sets.  We check whether I is an element of itself.

Q: 4. Is the set Q² x {a,b,c} countable or uncountable? You must justify your answer with a proof.

A: Given : ℚ2×a , b ,c  To Prove : ℚ2×a , b ,c  countable or uncountable

Q: If D =diag(λ1,λ2,...,λn), show that for every positive integer k, Dk =diag(λk 1,λk 2,...,λk n).

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Q: Suppose that A and B are sets. A is a subset of X and B is also a subset of X. Please teach me how…

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A: Let a, b and c be integers.

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Q: 4. Let D (d,}neN be a set of real numbers satisfying d,41 2 d, +1 for all ne N. Prove that D is an…

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A: {1, 3, 5}

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A: Ans : true , because

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Q: 1) If a and b are given integers, not both zero,show that the set T {ax+by: x,y are integers} is the…

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A: the objective is to solve the above questions.

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Q: 7. Consider the following subsets of R: A=[0,1] and K={all the (positive) prime numbers} Which set…

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Q: Let x and y be integers. Prove by the contrapositive that if x2 - y2 is even, the x + y is even

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Q: 1. Let A = {m, 2m,3m,...} where meP; that is, Aconsists of the positive integers of m. Find: a) A,…

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Q: (3.2) Indicate which of the following sets are compact in X. (a) C = {4,n € N}, X = R (b) C = Qn (0,…

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Q: Prove that if A and B are sets, then ( A ∩ B ) ∪ ( A ∩ B` ) = A

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Q: (8)Explain why: (a) Every rational number is an algebraic number (b) Z is not an open set (c) The…

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Q: For any positive integer n and r = 1,2, ...,n, we have G) = (","') + C" - }). True O False

A: The expression nr can be presented in the factorial form as follows. nr=n!(n-r)!r! The expression…

Q: disprove the following statement. The sum of an integer and its cube such that x + x° is

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Q: Let n and r be non-negative integers such that r <n. Then, C + "C , = **C,

A: nCr=n!r!(n-r)!

Q: b. H = {x E G|x" = e} for a fixed positive integer n.

A: Given set in part B

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Q: (c) Show that the set {r EN | x² % 4 = 1} is equal to the set {EN | x² % 8 = 1}.

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Q: Show that the set EE is: (ii). Closed iff E¯=E

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A: The Pigeonhole Principle:  If there are p+1 or more pigeons, need to distribute among p pigeonholes,…

Q: 1. Show that the set of integers S = {a² + b2 : a, b € Z} that can be respresented as the sum of two…

A: See the details below.

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Q: Show that 1 (N, <) is well-ordered. 1 2 (Z, <) is not well-ordered. 3 (R, <) is not well-ordered.

A: Please send questions as different questions. We cannot do more than 3 at a time.   A relation ≤ on…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.