3. Show that for any real number x, if x2 is odd, then x is odd.

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: Prove that for every real number b, where b is not equal to 0,= b.

A: Given

Q: 3. Find the number that provides a counterexample to show that the given statement is false. a) For…

A: Hi, thank you for the question. Since you have posted multiple questions with multiple sub-parts, as…

Q: Prove by contrapositive: if x is odd then x²+3x+5 is odd

A: Concept: Contrapositive method: A conditional statement's contrapositive is built through negating…

Q: Suppose there is a real number r with the property that r + 1/r is anodd integer. Prove that r is…

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Q: Show by contraposition that for positive real numbers a and b, if ab - a - b ≥0 then a and b are…

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

A:

Q: 1. Prove that for every real number x, if |x-3| > 3 then x² > 6x. (Hint: Consider two cases)

A: Solution

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x+ 5y² is even

A: Suppose x and y are integers. Prove that if x and y are odd, then x+5y2 is even.

Q: If 40% of x is 300 then find x

A: Given,40% of x is 300then, x = ?

Q: Prove or disprove if x2-1 is odd, then x is even

A:

Q: Show that the Archimedean Property is a consequence of the assertion that every interval (a,b)…

A: Archimedian property : For every positive real number c there exist a natural number n such that…

Q: Find a number that provides a counterexample to show that a given statement is false, For all number…

A:

Q: Prove that for every real numbers a, b, and c, where b a is not equal to zero, if a +b = c, then=+1

A: Given,            For every real numbers a,b and c, where b is not equal to zero.            if…

Q: 10. (a) Prove: If x and y are real numbers with x < y, then there are infinitely many rational…

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

A:

Q: Show that if f1(x) isΘ(g1(x)), f2(x) isΘ(g2(x)), and f2(x) ≠ 0 and g2(x) ≠ 0 for all real numbers x…

A:

Q: Find the value of x to make a counterexample to the conjecture. Conjecture: For all numbers x,…

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Q: Show that for any real numbers X and y. one of the following two case can happen: Either [x+Y = [×]…

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Q: Prove by contradiction that for any positive two real numbers, x and y, it x*y ≤50, then either x <…

A: Proof by contradiction Suppose to the contrary that if x*y ≤50 then x≥8and y ≥8, where x and y are…

Q: Suppose that k and n > k are fixed positive integers. Justify the identity k+1,

A: Given k and n≥k are fixed positive integers.

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

A:

Q: Prove that if x is an even integer, thenx^2−6x+ 3is odd

A: Any integer of the form 2m+1 is odd.

Q: x]+n x+n m m

A: By using definition of greatest integer function, For x∈ℝ we have, x=x+x, 0≤x&lt;1 where, x is…

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

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Q: If x and y are two positive real numbers, then there exists a positive integer n such that ny>x

A: Proof: Suppose there does not exist any positive integer n such that ny&gt;x then ny≤x   ∀n∈ℕ

Q: Use forward reasoning to show that if x is a nonzero real number, then x2 + > 2.

A: The detailed solution is as follows below:

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

A:

Q: Prove that x (x-2) is always a negative real number if x 1-Ai for some real number A.

A: The given number is  x(x - 2) x = 1 - Ai

Q: 2. Determine all values of x - y when x and y are integers such that x = 360 + y².

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Q: 2. Show that if a is an integer other than 0, then a) 1 divides a h) a divides 0

A:

Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

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

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Q: If b is a positive real number with b ≠1, then for all positive real numbers x and y we find

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Q: 3. Suppose a is an irrational positive real number. Prove that if x and y are integers and x # y…

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Q: 4. Prove that if m and n are positive integers and x is a real number, then

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Q: Prove that if x is real number then L2xd =LxJe bx+!]

A: We will prove the expression by two  steps taking real number x in two form.

Q: Suppose that x is a negative real number and that x < 1/x. Prove that x <-1.

A: Since we know for any real number a,b  If a &lt; b then -b &gt; -a

Q: Show that |x^n| = |x| ^n for every positive integer n and every real number x.

A:

Q: Let x, Y be positive real numbers. If x2 – y² =1, then x or y (or both) is not an integer.

A:

Q: Prove by contradiction that if a sum of three real numbers is less than 30, th at least one of the…

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Q: Prove that if æ is a positive real number, then

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Q: 4 If x is an integer, then xr >x.

A: See the detailed solution below.

Q: If x is any real number, then

A:

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y? is odd.

A: We have to solve given problem:

Q: h >x hɛIA

A:

Q: m above, prove that for every real number c>0, there is a 1/N < c.

A: In every real number c, there is integer N which is N &gt; c.

Q: Suppose also that P(X 2 8) = 3/4. Use Chebyshev's inequality to find a number u e R such that o u.…

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.