Let x be an element of the real numbers. Prove |x| is greater-than or equal to 0. State any theorems that are used in the proof.

Q: Let a and b be real numbers. Prove that | a b | = | a | | b | , b ≠ 0

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Q: 74. Prove that for any real numbers a and b, |b − a|=|a − b|.

A: Given; a and b are any real numbers Consider, L.H.S=b-aWe can write it as,=b+-a=-1-b+a=-1a-b…

Q: Determine the set of all real numbers x such that x-9 exists. x2-9

A: x-9x2-9

Q: Prove that the interval (0,1) ⊆ R is equinumerous with the interval (3,7), where R is the set of…

A: To establish a one to one  , onto (bijective map) between the open intervals (0,1) and (3,7)

Q: A(n) ______ ____________, denoted [a, b], consists of all real numbers x for which a ≤ x ≤ b.

A: It is known that any closed interval a, b can be represented as an inequality a≤x≤b.

Q: Given the domain and codomain of set of all real numbers, categorize the ff.: f (x) = 2x³ – 4…

A: Given - The domain and codomain is the set of real numbers. Function given is f(x)=2x3-4 which can…

Q: Define g: R→Z by g(x) = [x+x (g(x) = ceiling of x plus floor of x), where Ris the set of all reals…

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Q: Prove the following using the specified technique: Let x and y be two real numbers such that x+y is…

A: Consider the given information. Given that x and y be two real numbers such that x+y is rational. To…

Q: Prove the following by proof by contradiction: At least one of the 5 real numbers a, b, c, d, e is…

A: We will use basic properties  of inequality to proof this result by contradiction

Q: Prove. If x is a negative real number, there exists an integer k such that k >= x.

A: x is a negative real number then  x < 0 So that desired integer k = 0

Q: Prove that for all real numbers r and s, if r is rational, r≠0, and s is irrational, then rs is…

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Q: For any number a, prove that |-a|=|a|

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Q: If x and y are real numbers and x>1, prove that there exists a positive integer n such that xn > y

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Q: R.4. Prove that the set of real numbers in the interval (0, 1) which have a terminating decimal…

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Q: Prove that for all positive real a we have 22 +5 20x4 2 4/5. 24 x2 +5

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Q: Disprove the statement: There is a real number x such that xº +x* +1 = 2x².

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Q: Let x be a real number. Prove the following statement: for each e > 0, there exists a rational…

A: Given  that x be a real number. And we have to prove the following statement: for each ε>0, there…

Q: Find all values of x such that x² 2+ x + 2! = 0. 3! 4! If no such values of x exist, prove this. + +

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Q: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

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Q: Suppose that a is a positive real number not equal to 1, and that r and s are positive real numbers.…

A: To prove that; rlogαs=slogαr  We have to show that the left hand side of the equation equal to the…

Q: Prove : there exists a real number x such that x^2=3.

A: Let us assume a function f(x)=x2 For x=1, f(1)=12=1  For x=2, f(2)=22=4

Q: Use proof by contradiction to show that there is no positive real number x such that (x-2)/x ≥…

A: In an inequality, if a positive number is multiplied then there is no change of sign of inequality.…

Q: Prove that there exists irrational numbers r and y such that x" is

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Q: Assume B is a closed subset of the real numbers. Prove that Bc = R−B is an open set.

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Q: Let c, a, B be nonzero real numbers such thatc + B. Find a given that

A: This question is taken from the series and sequences in which we have to find the value of the alpha…

Q: 2. For all real numbers a, b and c, prove that if a <b <c then a+b+c a < 3 < C.

A: given that a, b, c are real numbers we have to prove that if      a&lt;b&lt;c    then…

Q: Let {x1, x2,..., xn} C R be a subset of real numbers. Prove that (x1 + x2 + . + an)² < n(x² + x +·.…

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Q: show that there doesnt exist a rational numbert such that t2 = 3

A: According to the given information, it is required to show that there does not exist any rational…

Q: Prove that there exists a unique positive real number, x, such that x = x^2.

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Q: prove for every positive real number x that 1+1/x2>=1/x+1/x3.

A: To prove: 1+1x2≥1x+1x3; for everypositive real number x.

Q: The set of all real numbers x for which x24x+2+x>0 is

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Q: Suppose x is a real number. Prove that if x 2, then there is a real number y such that x = (2y + 1)/…

A: Domain of a Function The domain of a function explains what values the independent variable, X, can…

Q: Prove: |a| &gt;= 0  for all a's in the reals.

A: It is known that the absolute value is the distance of the point or number from the origin.

Q: Let c, a, ß be nonzero real numbers such that c + B. Find a given that 52n-3 ac2 17 cn–4 n=3 с — в O…

A: According to question given that ∑n=3∞1752n-3cn-4=αc2c-β

Q: If x and y are real numbers and x>1, prove that there exists a positive integer n such that x" > y

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Q: Let x be a real number. Prove by contrapositive that if x + 5x = 40, then x < 3.

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Q: For every real number x+1, n-1 1+x+x² +x' +. x´+x' +...+x" for all neN. 1-x

A: The question has a mistake , as in the R.H.S the power of x must be (n+1).

Q: Theorem: for every real number x, x s |x]. Proof: If x 2 0, by definition |x| = x. Thus |x| > x. If…

A: Contradiction proof : Assuming opposite of our conclusion we show that our given conditions are…

Q: Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute…

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Q: Let x be a real number. Prove by contrapositive that if æ + 7x = 90, then x < 4.

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Q: Let x be an irrational number. Prove that there exists an irrational number y such that xy is a…

A: Let x be an irrational number. The main objective is to prove that there exists an irrational number…

Q: Show that if f1(x) is O(g1(x)), f2(x) is O(92(x)), and f2(x) # 0 and g2(x) # 0 for all real numbers…

A: Big theta notation is nothing but an order notation that is used to compare the the growth rates or…

Q: Let c, a, ß be nonzero real numbers such that c + B. Find a given that 52n-3 19 cn-4 ac² с — В n=3 O…

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Q: Let a be a positive real number. Prove that if a a

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Q: Prove that for all real numbers x, if x is not an integer, then [r – 1] = [x] – 2.

A: We have to prove x−1=x−2 for all real number x, if x is not an integer. Use definition:- x is the…

Q: . Prove that |x|+ |x + ;] = |2x| for any real number .

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Q: Use proof by contradiction to prove the following statement: For all real numbers x and y, if x/y is…

A: To prove the given statement using contradiction, we will first assume that x is rational.

Statements and Logic

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