Prove or provide a counterexample of the following statements: (a) If c and d are perfect squares, then cd is a perfect square. (b) If cd is a perfect square and c+ d, then c and d are perfect squares. (c) If c and d are perfect squares such that c> d, and x? = c and y? = d, then r> y. (Assume r and y are integers.)

Q: . Rewrite the equality of ideals (r+ y/2) = (3 – 4/2) in Z[/2] in terms of elements.

A: We know that, A subring  B of a ring R is called a ideal of R if for every r∈R and for every b∈B,…

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Q: In real-number multiplication, if uv = uw and u # 0, we can cancel u and conclude that v = w. Does…

A: To explain: either v=w or give counter example.

Q: b) Use an English proof to prove that if integers m and n are perfect squares, then m*n must also be…

A: Consider the provided question, Given, m and n are both perfect square. We need to prove m*n is also…

Q: Derive the Quadratic Formula, by selecting the appropriate pulldowns for J, K, L, M, N, P, and Q.…

A: Explanation of the answer is as follows

Q: 2. Prove each of the following statements: (a) There are infinite perfect squares. (b) There is no…

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Q: Prove that a perfect square must be of the form 5n or 5n + 1.

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Q: show that Q has infinitely many distinct quadratic extensions in C

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Q: Prove that |z₁+z₂²= |z₁|²+ z₂², if z₁/z2 is purely imaginary.

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Q: II. Prove the following using Direct Proof. . If x and y are perfect squares, then xy is also a…

A: In mathematics and logic, a Direct Proof is a way of showing the truth or falsehood of a given…

Q: Prove that a perfect square must be of the form 5n or 5n ± 1.

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Q: Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i].

A: We have to show that 2 is equal to the product of a unit and the square of an irreducible in Zi. The…

Q: Given that Z and w are. complex rumbers in C, which the following Statements Is true? B. 7 =Z if and…

A: As per the policy we are solving 3 subparts. Please specify which subpart is to be solved. Given : z…

Q: Prove that if x is even and y is even, then the product of x and y is divisible by 4.

A: We use the Division Algorithm to prove the given statement.

Q: Let vị = V2 = V3 V4 and -[1} Confirm that p = V1 +v2 + ¿v3 + ¿V4 and vị – V2 + V3 – V4 = 0. Jse the…

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Q: Q6. A. If gcd( m,91) = gcd( n, 91) = 1 then show that 91|m37 – n37 %3D %3D B. If m is a perfect…

A: The given problem is to prove the following statements. We have to use basic identities of number…

Q: 1 1 prove that 1+2@ 2+W 1+w If w is an imaginary cube root of unity,

A: If ω is an imaginary cube root of unity then (1) 1+ω+ω2=0(2) ω3=1

Q: Prove that if A ∈ M 2×2(F) is upper triangular, then det(A) equals the product of the diagonal…

A: Let the matrix A be the upper triangular matrix as follows.

Q: (k) If x, y, and z are integers and xylz, then x|z and ylz.

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Q: 3. Prove that ur2 +S is irrational. (Hint you may Use the Theorem y.71 which states that ve is…

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Q: Determine the least-norm solution to the following: X1 з01-11 x2 -1 20 -3] x3 –12 X4 274 77 4 12 11…

A: Step:- 1 Given system is 301-1-120-3x1x2x3x4=-124 of the form Ax=b Here, A=301-1-120-3, x=x1x2x3x4…

Q: Use proof by contradiction to show that no integers  y and z exist for which 24y + 12z = 1.

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Q: R.2. Let S represent the set of positive integers that are perfect squares, and let C represent the…

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Q: 2. Find the (infinity norm) condition number of 2.01 1 (b) A = 3 6.

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Q: Given that 22 = Z4, prove that Zl= 23. 1/2 3 st at least three facts you may use to complete this…

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Q: 20.* a. Let P(z) = 1 + 2z + 3z² +.. of P(2) are inside the unit disc. + nz"-1, By considering (1 –…

A: Proved by using Rouches theorem from complex analysis.

Q: Prove that 1 (c) 2F₁ (a, b; c; z)

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Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: 3. If AB and CD bisect each other at O, prove that AD || BC. Use two-column proof. A B

A: If AB and CD bisect each other at O, prove that AD∥BC.To use two‐column proof.

Q: Show that if n e Z is a perfect square such that 3|n, then 9|n.

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Q: Prove that V2 is irrational.

A: We will prove by the method of contradiction. We will first assume the number to be rational and…

Q: Prove that if S is dense in the real numbers, and I = (a, b) is an open interval, then I contains…

A: Prove that if S is dense in the real numbers, and I = (a, b) is an open interval, then I contains…

Q: Let z= a + ib is it true or false that |z²| = (Re(z)) + (Im(z)) where Re(z) is the real part of z…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2. Show that if x and y are perfect squares (that is, x = m? and y = n2, where m and n are…

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Q: Prove that every perfect square is either a multiple of 5 or of the form 5j - 1 or 5j + 1.

A: Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap −…

Q: Show directly that for p a prime integer, p can be written as p=(a)^2 - d(b)^2 then d is a perfect…

A: From the given statement, p be the prime factor and p can be written as p = a2 – db2.To show: d is a…

Q: if n is a perfect square then n2 + 2n is not a perfect square. prove this

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Q: Prove the given statement: The sum of the squares of two odd integers cannot be a perfect square.

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Q: b) find the meet and join. of the ZeYO - One maFrices A = 1 10 %3D 111 010 and B = 000 [101 1111]

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Q: Prove that |z₁+z₂²2 = |z₁|² + 12₂², if z₁/z₂ is purely imaginary.

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Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(AB) =…

A: Suppose A and B are bounded and nonempty subsets of real numbers and λ∈R. We need to prove that…

Q: 7. In quadrilateral FGHJ, FH and GJ bisect each other and FH = GJ. FG JH FJ GH What additional…

A: FH and GJ are bisector of each other.

Q: 40. Let d be an integer that is not a perfect square. Show that Q(Vd) (a+ bVdla, be Qj is a subfield…

A: To show that Qd is a subfield of C. By subfield test we need to only show that; For x,y∈Qd, y not…

Q: 6. Prove that |x.y| = |x|.|y|, where x and y are real numbers.

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Q: (4) Prove that if s and t are odd perfect squares, s + t is not a perfect square. Use contradiction…

A: A square of a number is the product of a number with the number itself . A perfect square is the…

Q: Prove that |z₁+z₂|²= |z₁|² + z₂², if z₁/z₂ is purely imaginary.

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Q: 3.9. (a). Prove that z is either real or purely imaginary if and only if (2)2 = 22. (b). Prove that…

A: The given problem is to prove the following complex identities.

Q: Find bases for the null space and row space of A. 3 4 6. -10 -8 -4 -16 A = -1 -1 0 -2 3 4 7 13 14

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Q: Let A = {n ℤ | n = 5r for some integer r}and B = {m ℤ | m = 20s for some integer s}.Determine which…

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Q: Prove that if m is a positive integer and x is a real number, then [m2] = [=] + z++ + 2 + |x + т — 1…

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.