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Let H be a subgroup of G. Let a, b ∈ G. Prove: aH = bH if and only if a⁻¹b ∈ H. b) If aH = bH, then Ha⁻¹ = Hb⁻¹.

          Let H be a subgroup of G. Let a, b ∈ G. Prove: aH = bH if and only if a⁻¹b ∈ H. b) If aH = bH, then Ha⁻¹ = Hb⁻¹.
        

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A Book of Abstract Algebra
A Book of Abstract Algebra
Charles C. Pinter 1982 Edition
Chapter 5
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Let H be a subgroup of G. Let a, b ∈ G. Prove: aH = bH if and only if a⁻¹b ∈ H. b) If aH = bH, then Ha⁻¹ = Hb⁻¹.
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Transcript

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00:01 In the question they given raymond sum approximation of integral 1 to 5 f of x bx.
00:08 Here we have to find out width of each subinterval.
00:12 In the question they given a graph.
00:14 First we have to write were a equal to 1, b equal to 5 and n equal to 4.
00:24 Then the width of each interval is, the formula of width of each interval is del x equal to b minus a divided by n.
00:47 Here we know the values of b, a and n.
00:51 We have to substitute in this formula 5 minus 4 divided by 4.
00:57 Here a value is 1.
01:00 So 5 minus 1 divided by 4 that equal to 4 by 4 that equal to 1.
01:07 That equal to 1.
01:08 Here the del x value is 1.
01:15 Then next we have to write midpoint ram and sun is defined as integral.
01:36 In the question they given 1 integral.
01:39 We have to write that integral integral.
01:43 1 to the power of 5 f of x d x.
01:48 Here f of x value is tel x, f of x naught plus x1 divided by 2 plus f of x1 plus x0 divided by 2 x2 by 2x2 by 2 plus f of x2 plus x2 plus x3 divided by 2 2 plus here we have to write this continuous up to x4 x3 plus x4 by 2 then its graph is now we have to draw that graph then its graph is x not equal to 1 x2 equal to 2 x3 equal to 3 x 4 equal to 4 we have to have to draw that graph.
03:06 Here n equal to 4...
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