Question

Pe multimea R se defineste x ° y = x + y + 2, asociativa.
a) Determinati simetricul lui 5 in raport cu legea ,,° ".
b) Determinati x apartine R , x ° x ° x ° x ° x = 3.​

Answer 1

Explicație pas cu pas:

a)

$x \circ e = e \circ x = x$

$\begin{cases}x \circ e = x + e + 2 = x \\e \circ x = e + x + 2 = x \end{cases} \iff \begin{cases}e + 2 = 0 \\e + 2 = 0 \end{cases}$

$\begin{cases} e = - 2 \\ e = - 2 \end{cases} \implies e = - 2$

$x \circ x' = x' \circ x = e$

$\begin{cases}5 \circ x' = 5 + x' + 2 = -2 \\x' \circ 5 = x' + 5 + 2 = -2 \end{cases} \iff \begin{cases}x' + 7 = - 2 \\x' + 7 = - 2 \end{cases}$

$\begin{cases}x' = - 9 \\x' = - 9 \end{cases} \implies \bf x' = - 9$

b)

$x \circ x \circ x \circ x \circ x = 3$

$x \circ x = x + x + 2 = 2x + 2$

$x \circ x \circ x = x \circ (x \circ x) = x \circ (2x + 2) = x + (2x + 2) + 2 = 3x + 4$

$x \circ x \circ x \circ x = x \circ (x \circ x \circ x) = x \circ (3x + 4) = x + (3x + 4) + 2 = 4x + 6$

$x \circ x \circ x \circ x \circ x = x \circ (x \circ x \circ x \circ x) = x \circ (4x + 6) = x + (4x + 6) + 2 = 5x + 8$

$5x + 8 = 3$

$5x = -5 \implies \bf x = -1$