Let a, b e C. (i) Prove that if |a| <1 and |6| < 1, then a - b < 1. 1- āb

Q: a – b -| < 1, if|a| < 1, and |b| < 1. 1- āb - Show that|-

A: From the given information. For , then show that .

Q: Prove: | a | - | b | &lt;= | a + b |

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Q: Prove or disprove the statement: For all x E R, |x + 1| + |x – 1| > 2.

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Q: 2. Show that for any real number c, there is exactly one integer in the interval (c,c + 1].

A: As per guidelines, I can do only first question.

Q: (a) Explain with cases why, if a is a real number,  then a ≤ |a|. (b) Explain, if, for any real…

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Q: A and B are finite sets Prove that |AxB| = |A||B|. %3D

A: Let A=1, 2, 3, B=a, b, c

Q: Let x1, x2,...,n be real numbers. Prove that |X1 + X2 + ··· + Xn] ≤ |X1| + |X2| + · · · + |xn|·

A: The objective of this question is to prove the triangle inequality for a sequence of real numbers.…

Q: If r and s are real numbers and r < s, then there exists a positive integer t such that r+t = s.…

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Q: if |z² + 1| = 2| z+1|  z e C then | z|  <= sqrt*7 Prove.

A: Given  z2+1=2z+1 , z∈ℂ            (i) Now,   z2+1=2z+1z2+12=4z+12 ,…

Q: If 0 < a < b, prove that 0 < a2 < b2 and 0 < √a < √b

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Q: Let a be a positive real number and n be a natural number. Define S = {r €R:r20 and a" a, show that…

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Q: |°z\+|'z| |*z\-|°z| Z1 +z2 Z 2 +|z. Z3 + z, Prove: If Fs|#/z,) show that ' 4 VI

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Q: Question 1. 1. Use the operations in o-algebra, such as union, intersection, difference, countable…

A: Using the operations in σ-algebra generate the following intervals from intervals of the form a,∞,…

Q: (B) Prove that (Q,|·|) is not complete.

A: A metric space (X,d) is said to be complete iff every Cauchy sequence in X converges to a point in…

Q: z = -2 verify that |7 - 2| < |||| - |||

A: Given that y¯=−310 and z¯=1−21. We have to verify y¯⋅z¯≤y¯⋅z¯. Now y¯⋅z¯=−310⋅1−21=−5 Hence, y¯⋅z¯=5…

Q: Let a, b, d E Z. Prove using only the definition of divisibility (that is, no other lemmas or…

A: Solution:- It is given that ,    a , b , d ∈ℤ then , we have to prove that  If     d /  a   and  d /…

Q: Show that if x,y,z,w are real numbers such that x < y and z ≤w, then x + z < y + w.

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Q: Prove that |(a, b)| = |(c, d)| for any numbers a, b, c, d where a < b and c < d. Proof: Prove that |…

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Q: Prove the following statement in three different ways, using a direct proof, a contrapositive proof…

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Q: (a) Using Proposition 1.3.2 and/or the definition of absolute value, prove that if b ‡ 0, 1 1|2|=…

A: The proposition is given by :  For any two real numbers  x and y  we have       x2=x ,  x2=x2  ,…

Q: Discrete Mathematics: Prove that for all real numbers x, y, and z, if x + y + z ≥ 3, then either x ≥…

A: We will prove by method of contradiction

Q: Prove: For c &gt;= 0 , | a | &lt;= c if and only if -c &lt;= a &lt;= c

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Q: Proof the following statement in detail: ". Let x ∈ R. Prove that if 3x4 + 1 ≤ x7 + x3, then x >…

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Q: 1 1 1 (c) Find simple descriptions for each of the sets U • U [ ₁ - ² x + ¹) and № [² - ² + ¹) X- n…

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Q: 1. Show that if P(A) > 0, then P(AB|A) > P(AB|AUB)

A: Show that if P(A) &gt; 0 thenP(AB|A) &gt; P(AB|AUB)

Q: Question 1. (a) Prove that for z, w = C, Re (zw) = |z||w| if and only if z = sw for some positive…

A: a)  suppose that the complex numbers are   z=a+ib and w=c+id .  Then we see that w¯= c-id  we see…

Q: 1.5 Prove that e-2=| 0.

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Q: Prove the following statements: i. {x €R : |x – 4| = 2|x| – 2} ={-6,2}

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