تحديد مجموعات الأعداد حسب شرط بالقيمة المطلقة
\( A = \left\{ x \in \mathbb{Z} \mid |x| = 5 \right\} \)
\( |x| = 5 \)
\( \Rightarrow x = 5 \) أو \( x = -5 \)
\( A = \left\{ -5 ; 5 \right\} \)
\( B = \left\{ x \in \mathbb{Z} \mid |x| = -6 \right\} \)
\( |x| = -6 \) غير ممكن
لأن القيمة المطلقة لا تكون سالبة
\( B = \emptyset \)
\( C = \left\{ x \in \mathbb{Z} \mid |x| = 4.5 \right\} \)
\( |x| = 4.5 \)
لكن \( 4.5 \notin \mathbb{Z} \)
\( C = \emptyset \)
\( D = \left\{ x \in \mathbb{Z}_- \mid |x| = 2 \right\} \)
\( |x| = 2 \Rightarrow x = \pm 2 \)
لكن \( x \in \mathbb{Z}_- \Rightarrow x = -2 \)
\( D = \left\{ -2 \right\} \)
\( E = \left\{ x \in \mathbb{Z}_+ \mid |x| = 11 \right\} \)
\( |x| = 11 \Rightarrow x = \pm 11 \)
لكن \( x \in \mathbb{Z}_+ \Rightarrow x = 11 \)
\( E = \left\{ 11 \right\} \)
\( F = \left\{ x \in \mathbb{N} \mid |x| = \frac{3}{7} \right\} \)
\( \frac{3}{7} \notin \mathbb{N} \)
\( F = \emptyset \)
\( G = \left\{ x \in \mathbb{Z} \mid |x| = \frac{15}{3} \right\} \)
\( \frac{15}{3} = 5 \)
\( \Rightarrow x = \pm 5 \)
\( G = \left\{ -5 ; 5 \right\} \)
\( H = \left\{ x \in \mathbb{Z}_- \mid |x| = -7 \right\} \)
\( |x| = -7 \) غير ممكن
\( H = \emptyset \)
\( I = \left\{ x \in \mathbb{Z} \mid |x| = x \right\} \)
القيمة المطلقة تساوي العدد فقط عندما \( x \geq 0 \)
\( I = \left\{ x \in \mathbb{Z} \mid x \geq 0 \right\} = \mathbb{Z}_+ \)
\( J = \left\{ x \in \mathbb{Z} \mid |x| = -x \right\} \)
القيمة المطلقة تساوي \( -x \) عندما \( x \leq 0 \)
\( J = \left\{ x \in \mathbb{Z} \mid x \leq 0 \right\} = \mathbb{Z}_- \)
\( K = \left\{ x \in \mathbb{Z}_- \mid |x| = x \right\} \)
\( x \in \mathbb{Z}_- \Rightarrow x < 0 \)
\( |x| = -x \ne x \)
\( K = \emptyset \)
\( L = \left\{ x \in \mathbb{Z}_+ \mid |x| = -x \right\} \)
\( x \in \mathbb{Z}_+ \Rightarrow x \geq 0 \)
لكن \( |x| = x \ne -x \)
\( L = \emptyset \)
\( M = \left\{ x \in \mathbb{Z} \mid |x| < 4 \right\} \)
\( |x| < 4 \Rightarrow x \in \left\{ -3, -2, -1, 0, 1, 2, 3 \right\} \)
\( M = \left\{ -3, -2, -1, 0, 1, 2, 3 \right\} \)
\( N = \left\{ x \in \mathbb{Z}_- \mid |x| < 3 \right\} \)
\( x \in \mathbb{Z}_- \Rightarrow x < 0 \)
\( |x| < 3 \Rightarrow x \in \left\{ -2, -1 \right\} \)
\( N = \left\{ -2, -1 \right\} \)
\( P = \left\{ x \in \mathbb{Z} \mid |x| = |-(-7)| \right\} \)
\( |-(-7)| = |7| = 7 \)
\( \Rightarrow x = \pm 7 \)
\( P = \left\{ -7, 7 \right\} \)