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A∈ S ais in the set S S= T the sets S and T are equal, ie, every element of S is in T and every element of T is in S S⊆ T the set Sis a subset of the set T, ie, every element of Sis also an element of T ∃a∈ S P(a) there exists an ain S for which the property P holds ∀x∈ S P(a) property P holds for every element in SThe Largest Database for the Root Solutions on the InternetT t B A n C A G ݁A X m { h A } X c n ̑ V b s O T C g N u } E F u X g A B E F u l ɂ͊ L I o ^ A ȒP ɓo ^ o ܂ B T t B p i A T tDVD A X P g { h p i A X m { hDVD ̒ʔ̐ X Surf Item T
9 Fourier Transform Properties Solutions to Recommended Problems S91 The Fourier transform of x(t) is X(w) = x(t)e jw dt = fe t/2 u(t)e dt (S911) Since u(t)CVmVaV å edZibVa, mhd =dY ^ba X ВX^Zi, mhd imc^Zdgh^Yad fadgh^ X bd_ \^c^ Hd edZc å edcåa, mhd =dY c YdXdf^a `dc`fhcd dWd bc mr naV d hdb, mhd cVghVad Xfbå edZa^hrgå X Wda n^fd`db bVgnhVW edcVc^b dh`fdXc^å d bda^hX cV ^cqk åq`Vk, `dhdfd Ic dh`fqXVa bc X hmc^ bcdY^k ahPART 1 MODULE 2 SET INTERSECTION, SET UNION, SET COMPLEMENT SUMMARY The intersection of two sets denotes the elements that the sets have in common, or the "overlap" of the two sets S ∩ T = {xx∈ S and x∈ T} The union of two sets merges the two sets into one "larger" set S ∪ T = {xx ∈ S or x ∈ T}
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(b)Generalise the result from part (a) to T Fn!F (c)State and prove an analogous result for T Fn!Fm Solution (a)By linearity, T(x;y;z) = xT(1;0;0) yT(0;1;0) zT(0;0;1);F(x) hasa value equal to f(c) = ∫b a f(x)dx b a Multiplying bothsides by b a proves the result 4The first fundamental theorem of integral calculus We are now in a position to prove our first major result about the definite integral The result concerns the socalled area function F(x) = ∫ x a f(t)dt and its derivative with respect to x
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