1. For each of the statement below, decide whether it is true or false, and prove your claim. (a) For all a, b R, if a € Q \ {0} and b & Q then % & Q. (b) For all x, y € R, if x is irrational and y is positive and irrational, then x + √y is irrational.

Q: X7. We know that given any real numbers x1 and x2, if x1x2 = 0, then x1 = 0 or x2 = 0. Since there…

A: (b) given any real number x1 , x2 , x3 , x4 claim- if x1x2x3x4=0 then x1=0 , x2=0 , x3=0 , x4=0

Q: Which x satisfies both |x − 3| < 2 and |x − 5| < 1?

A: The inequalities are:  |x − 3| < 2 and |x − 5| < 1 Then we get,…

Q: a Prove that < 16 – a| for each a, b e R. - V62 + 1 Va? + 1

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Q: (a) For every integer n, the number n² — n must be even. (b) For all real numbers a, a² < a¹. (c)…

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Q: from x = 2 to x = 10 and x; is any value of x in Ex¡. Express lim s as a definite integral. n-x n D…

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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 which real numbers x do the following equations hold: (a) Vx + V2x – 1+ Vx – V2x – 1 = V 2а — 1…

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Q: Show that if x > 0 and |x² – 9| < 2, then |x - 3| < 1.

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Q: Consider the operation (#) for all real numbers which is defined as follows: for real numbers a and…

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Q: (c) Prove that r| +11 36 T-1 2 +36 for all real I.

A: We have to prove that the following inequality holds: x-11x2+36≤x+1136 for all real x. We will use…

Q: If P(x) is the statement x ≥ 11, Q (x) is the statement x ≤ -2 and R (x) is the statement (x - 3)(x…

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Q: Use the laws of logical equivalence to prove the equivalence below. A law must be stated as the…

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Q: Consider the sentences Vx3yP(x, y) and 3yVxP(x, y). Are these sentences true or false under the…

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Q: (a) For all a, b E R, if a E Q \ {0} and b & Q then ; ¢ Q. (b) For all x, Y E R, if x is irrational…

A: a) Given: a∈ℚ\ 0, b∉ℚ To prove: ab∉ℚ b)  Given: x∈ℝ-ℚ, y∈ℝ-ℚ+  To prove: x+y∈ℝ\ℚ Rational Number: A…

Q: 2. |a + ib|= √a² + b² O True O false

A: This question is based on modulus of complex number.

Q: hat p (A) =0.3 and P (B) = 0.2 find P (A or B) given that (a) A and B are mutually exclusive, (b) A…

A: p (A) =0.3 and P (B) = 0.2  (a) A and B are mutually exclusive P (A or B) = P(A) + P(B)…

Q: 1. (P16, Page 11,.. ) Use the Method of Proof by Contradiction to prove that 3 ¢ Q. You may use the…

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Q: (a) Suppose R be the set of real numbers. Given A = {x €R|-7 5}. Find: i. AUC ii. (A UB) NC iii. AA…

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Q: Let P (A) = 0.4 and P (A ∪ B) = 0.6. For what value of P (B) are A and B independent?

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Q: For a real number a, if a + 0 then a x a E R+.

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Q: Use Contrapositive method to proof the following statement. (a) Suppose a E Z, if a? + 5a is even,…

A: Given: using contrapositive method

Q: Assume the following three statements are true. p: For all real numbers x and y, if xy is…

A: We are given that statements p,  q and r are true. We need to prove that 2a-1 or a2+3 is…

Q: Let v = Find [-13]; ®= { [4]· [H] }; × = {[3]· B]}-· B C a. [v]: b. [v]: c. [B]: d. [B][v]: e. Are…

A: (.) Given,(.) Coordinate vector:  Let  be a basis of a vector space . Then coordinate vector of…

Q: 3. Find all r so that r-1+r-2| < 5.

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Q: X7. We know that given any real numbers x1 and x2, if x1X2 0, then x1 0 or x2 = 0. Since there are…

A: Given statement is P2 where P2: for any two given real numbers x1 and x2, if x1x2=0, then x1=0 or…

Q: b. {r|x € N and 7 <x< 14}

A: Given,  x|x∈ℕ and 7≤x≤14

Q: ) Recall that r > 0 for every positive real number r. Prove that if a and b are positive real…

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Q: If 0<x<4 then a < 6x + 2<b   a= b=

A: In this question, we need to find the values of a and b.

Q: For each of the statement below, decide whether it is true or false, and prove your claim. (a) For…

A: Sum , difference, product and division of rational numbers is rational

Q: Let x Ry be defined as follows: Ry means that 2 | (x - y). Select the true statements 6R3 10R5 6R6…

A: We have to select true statements using given data.

Q: Let P(x) denote the statement "x < 4." P(4) is A) True False

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Q: If xER then |1– x| = |x+1|- True O False

A: We have check given statement is true or not