An element x is in A ∪ B if, and only if, _____________________.

Q: If a, b. and c are integers such that a|b and c|b. then ac|b.

A: False

Q: 10) Let a and b be positive integers. Prove that (a, b) = a if and only if a divides b.

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Q: (a) A C B if and only if A U B = B. (b) A C B if and only if AN B = A. (c) A C B if and only if Bº C…

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Q: Prove by Algebra of Proposition, the following relation is a contradiction [(B v A v (B ^ C)] ^ ¬(A…

A: By using laws of logical equivalence we solve the given relation as follows :

Q: If a, b, and c are integers such that a|b and c|b. then ac|b. ylg

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Q: (2c) Prove or disprove the following statement: “ A UBCANB for any A and B "

A:

Q: If a, b, and c are integers such that a|b and c\b, then ac|b. а,

A: To check the given statement true or false lets take look a couple of example as follows : Take a=2…

Q: Let B(x) `x is not beautiful", V(x): `x is not a village in Lebanon.", and domain of x be dll…

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Q: Prove: For all integers a,b and c, if a|b and a|c, then a|(3b–2c)

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Q: Prove the statement; a – c is a divisor of ab + cd=→a – c is a divisor of ad + bc.

A: We need to prove the statement; a – c is a divisor of ab + cd => a – c is a divisor of ad + bc.

Q: Given the statement: If the sum of two integers, m and n, is odd then one of the integers must be…

A:

Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: Prove by any method(contradiction/contrapositive or cases) Prove that if a and b are positive…

A: Prove that if a and b are positive integers such that a|b and b|a, then a=b Let a and b are positive…

Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

A:

Q: Prove the following statements: (a) If two integers are both odd, then their sum is even. (b) If two…

A: Consider two odd integer

Q: Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

A: It has been given to us that a, b, c are integers and a2 + b2 = c2.…

Q: 18. Prove the following proposition: For all integers a and b, if 3 divides (a? + b?), then 3…

A: Suppose 3 divides a2+b2 for all integers a and b To prove 3 divides a and 3 divides b Let us prove…

Q: 5 points) Use a proof of the contrapositive to prove the following statem For all integers a, b, if…

A: The contrapositive of a statements  " If p then q  '' is    " If not q then not p "

Q: Let a, b be integers. Prove the following statement by contrapositive: If a - 46 is ODD, then a is…

A:

Q: Prove: If a, b and c≠0 are integers, then a|b if and only if ac|bc.

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Q: Which of the following statements is true: For all integers a, b and c, if (a + b)|c and then a | c…

A: Here, we have to find the true statement from the given four statements. For this, we shall check…

Q: For all integers a, b and c, a|bc if and only if a|b and a|c True or False

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Q: all integers h, k, and with

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Q: The elements of P({a, b, {a, b}})= 10, * .where a and b are distinct elements true False O

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Q: Prove that B \ A = B if and only if A ∩ B = ∅.

A: We will use definition  of sets for proving following question

Q: a.) Let a and b are positive integers. Prove that (a,b) = 2(a/2, b/2).

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Q: Prove or Disprove: If x and y are integers and x - y is odd, then x is odd or y is odd

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Q: Prove or disapprove that if a|bc, where a,b,c are positive integers and a≠0, then a|b or a|c

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Q: If Ua = {(x1, x2, x3)" E R³|x1+ 2x2 + 3ax3 = a} is closed under addition, then a = 0. Prove that the…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: If a, b. and c are integers such that alb and c|b. then ac|b.

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Q: Prove by contradiction that for all integers a, b, and c, if a|b and a | c, then a |(b + c).

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Q: Provide a proof by contraposition of the statement below. Prove that for any integer æ, if x + x +1…

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Q: If there exist integers r and s such that ra + sb = d. then d = (a, b). hi O

A: Given: There exist a two integers r and s such that ra+sb=d. To prove : d=(a,b)

Q: Prove that if a and b are both positive, and a2 < b2 then a < b          (Use Indirect Proof)

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Q: Which of the following statements is true: For all integers a, b, and c, a | be if and only if a | b…

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Q: (1) Disprove the statement "For all integers a, b if a | b2 and a < b then a | b." by giving a…

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Q: Which of the following statements is true: For all integers a, b and e, if (a + b)lc and then a | c…

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Q: Show that if a,b, and c≠0 are integers, then a∣b if and only if ac∣bc.

A: We have to prove the following statement -  If a,b, and c≠0 are integers, then a∣b if and only if…

Q: Suppose A is open. Consider the following statements: (i) If finitely many points are removed from…

A: Given: A is open (i)If finitely many points are removed from A, then A is still open. (ii) If…

Q: Prove the existential statement given below. There are integers c and d such that 7c + 5d 1.

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Q: Prove: For all integers a, b, and c, if alb and alc then a (2b+3c)

A: prove is given below ....

Q: (a) Let a,b, and c be integers such that a | b and a | c, and let x and y be arbitrary integers.…

A: According to the guidelines I am answering first question only. a) a, b and c be integers such that…

Q: If there exist integers r and s such that ra + sb = d. then d = (a, b). %3D

A:

Q: For all integers a, b, and c, a | bc if and only if a | b and a | c. ect one: True False

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Q: linear algebra, 2.5 question 5

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Q: (b) Give the contrapositive of the following statement concerning positive integers n,a and b: If n…

A: The negation of [a≤√n or b≤√n] is : a &gt; √n and b &gt; √n And, the negation of, n = ab is, n ≠ ab…

Q: and

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Q: What is the converse of "If there is ice on the lake, then it is cold?"" If there is not ice on the…

A: converse of "If there is ice on the lake, then it is cold" is ? Converse means q--&gt;p