Let æ be a real number. Show that [3r] = [a] + æ + + 3

Q: Which numbers satisfy |a| = a? Which satisfy |a|=−a? What about |−a| = a?

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Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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Q: For a given real number t, express tant in terms of sect

A: For a given real number t, express tan(t) in terms of sec(t)

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: Label each of the following statements as either true or false.   Every decimal representation of a…

A: To label the given statement

Q: Suppose that the number 'a' has a property that for every natural number n, a ≤ 1/n. Prove that a ≤…

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Q: Prove that if x is a positive real number, then LV L]] = [V]

A: We have to prove the given equation for all positive real numbers x .

Q: (e) For any nonzero real number z, prove Σ''-1 1-z"

A: The objective is to prove that for any nonzero real number x, ∑i=1nx-1xi=1-x-n

Q: Prove the statement if it is true, and give a counterexample if it is not true. a. For every real…

A: Both the subparts are solved with explanation.

Q: For all real numbers a and b, |a + b| < ]a| + |bl. Select one: OTrue False

A: Answer

Q: Label each of the following statements as either true or false.   Every decimal representation of a…

A: Decimal representation of a real number that does not terminate but repeats are rational.

Q: Let a and b be integers. Show that if every integer that divides a also divides b, then a | b. (a…

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Q: If α and β are two real numbers, we say that β is equivalent to α if there are integers a, b, c, and…

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

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Q: Prove that the value of an expression (a - 1)(a - 3)(a - 4)(a - 6) + 10 is positive for every a…

A: To prove that   the value of an expression (a - 1)(a - 3)(a - 4)(a - 6) + 10 is positive for every a…

Q: Let c, a, B be nonzero real numbers such that c + B. Find a given that 00 192n-3 ac? >18 %3D ch-4 с…

A: To find a from the given summation.

Q: List the members of each sets. 1( x/x is a real numbers such as x^3= 27) = 2. (x/x is an integer…

A: We will find each elements of the set

Q: Let n be a real number and let e be a positive real umber. Determine whether there exists a 1 real…

A: Given n be a real number and ε be a positive real number.

Q: Every positive integer is the sum of at most two squares and a cube of nonnegative integers. O True…

A: consider 0  0=02+02+03 now  1 can be written as  1=02+02+13 now consider 2 2=12+02+13 now consider 3…

Q: Let a, b, and c be natural numbers. Prove a ∙ (b + c) = a ∙b + a ∙ c.

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Q: . Use the definition of e" to show that for any real numbers 0, 01, and 02. a. ee, = 0, +0,). b. e"…

A: We have to prove that  a.  eiθ1·eiθ2=eiθ1+θ2b.  e-iθ=1eiθ We know the definition from Euler's…

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: Use proof by contradiction to prove the statement. If a, b, and c are real numbers such that ac ≥ bc…

A: For the given condition assume that c &lt; 0. Now, find the relation between a and b with the…

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

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Q: Prove that if a is an irrational number, then a-1 is an irrational number.

A: Given:           a is an irrational number To prove:            a-1 is an irrational number

Q: let a and b be integers. Show: if a * b is odd, then a is odd and b is odd

A: A number is even, if it is divisible by 2. A number is odd, if it is not divisible by 2. The odd…

Q: If a is any integer, is a(a + 1) even or odd?

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Q: Prove that if m is a positive integer and r is a real number, then Lma) = l=] + -+ + + [-+ + r+ m

A: We are given a Inequality involving the sum of floor functions, and we have to prove that.

Q: Prove: Let a, b, and c be real numbers. If a &lt; b and c &gt; 0, then ac &lt; bc.

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Q: For any real numbers a and b: If a -b

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Q: Show that given any particular real number, such as I, there is NO next real number immediately…

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Q: Determine whether the following statement is true or false: 1 ¢ {x|x is a natural number}. 4

A: Given statement is 14∉x|x is a natural number. To determine whether statement is true or false.…

Q: Let a, b, c be distinct positive real numbers and nbe a natural number. If a, b, c are either in…

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Q: Show that for every real number ɛ 8 > 3ɛ. greater than zero, there exist a real number 8 such that

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Q: Give an example of numbers a and b such that |a + b| &lt; |a|+|b|.

A: To give an example of numbers a and b such that

Q: Use proof by contradiction to prove the statement.If a is a real number and 0 &lt; a &lt; 1, then a2…

A:

Q: For any integer m and any real number x, if x is not an integer, then [x]+[m- x] = m- 1.

A: Given that m is an integer and x is a real number which is not an integer. we know that for any real…

Q: Let c, a, B be nonzero real numbers such that c + B. Find a given that 72n-3 Σ17 00 ac? cn 4 c - B…

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Q: Let a be a positive real number. Evaluate So Va? – x²dx.

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Q: . Show that if r is a nonzero real number, k and m are distinct integers, and ax and am are any real…

A: Given system is Crk+kDrk=akCrm+mDrm=am also given that r is non-zero real number and k and m are…

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: Let a, b, and c be nonzero real numbers. Show, without direct evaluation, that 1 1 1 1 0. a bc ac ab

A: To show the given condition

Q: Let a and b be real numbers. Prove that, if a + b >= 100, then a>39 or b>60

A: Let a and b be two real numbers, a, b∈ℝ.

Q: Prove that if x is a real number, then [2r] = [r] + r+

A: Let us consider x=n+∈     (Because x is a real number) where n is integer part and ∈ is fractional…

Q: Use the definition of e to show that for any real numbers 0, 01, and 02, a. "": = M0, +0,) b. e" =…

A:

Q: Disprove this statement by giving a counterexample. For all nonzero real numbers, a, b, c, and d,…

A: Here a,b,c,d all are non zero real .So (a/b) and (c/d) are defined.  Let,  a = 1 , b = 2 , c = 3 , d…

Q: If x is any real number, then

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Q: . The operation * is defined over the set R of real numbers by a *b = a+b+2ab. What is the identity…

A: We will find out the required value.

Q: 6. Let a be a real number.

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Q: Let a and b be real numbers. Prove that if 4a+16b= 123, then a and b are not integers

A: Topic :- Equations In the given solution below 123 is not completely divisible by 4.