Propriety of two random variables

Question

Consider two random variables X and Y. Determine which of the following statements is true, justifying the answer and specifying any additional hypotheses necessary to make them true.
a) $E[X|X] + E [Y|Y] = X + Y$ ;
b) $E [X + Y| |X| = x] = x + Y$ ;
c) $E [X | |X|] = E [X|X]$.
d) $E [g(X)h(Y)|X] = g(X)E[h(Y)|X]$

Here is my attempt:
a)$E[X|X] + E[Y|Y]\overset{?}{=} X + Y$
Generally $E[X|X] = X$ so True
b)$E[X+Y||X|=x] \overset{?}{=} x + Y$
$= \int_R \int_R\mathrm(X+Y||X|=x)f(x,y)\,\mathrm{d}x\mathrm{d}y$
$= \int_R \mathrm Xf(x,y)\,\mathrm{d}x + \int_R \mathrm (Y||X|=x)\,\mathrm{d}y$
$= E(X) + Y \neq x + Y$ so False

c)$E[X||X|] \overset{?}{=} E[X|X]$
I know that $E(X|X) = X$, so $E[X||X|] \overset{?}{=} X$ (and i don't know exactly what to do at this point)

d)$E[g(X)h(Y)|X] \overset{?}{=} g(X)E[h(Y)|X]$
$= E[g(x)h(Y)|X = x]$
$= E[g(x)]E[h(Y)|x]$
$= g(x)E[h(Y)|X]$ so True

Thanks for helping me.

Answer 1

For part $b$ and $c$ consider that when given $\lvert X\rvert = x$, then $X$ may be either $-x$ or $x$ with some conditional probability, which evaluates the conditional expectation as:$$\mathsf E\big(X~\big\vert~ \lvert X\rvert\,{=}\,x\big)~=~x\,\mathsf P\big(X\,{=}\,x~\big\vert~\lvert X\rvert\,{=}\,x\big)-x\,\mathsf P\big(X\,{=}\,{-x}~\big\vert~\lvert X\rvert\,{=}\,x\big)$$