Monday, April 12, 2021

Prodhimi kartezian

Përkufizim: “Prodhim kartezian i dy bashkësive A dhe B quhet bashkësia C, e formuar nga të gjitha çiftet e radhitura, ku elementi i parë i çiftit është element i A dhe elementi i dytë është i B”.

Simbolikisht shënohet AxB dhe lexohet prodhimi kartezian i bashkësisë A me bashkësinë B.

Çiftet e radhita (a, b) dhe (b, a) nuk janë njësoj.

Shembull 1

Gjeni prodhimin kartezian AxB, nëse A = {3, 2, 5, 4} dhe $\displaystyle B=\left\{ 1,4,2,5, \right\}$

Zgjidhje

$\displaystyle AxB=\{\left( 3,1 \right),\left( 3,4 \right),\left( 3,2 \right),\left( 3,5 \right),$

$\displaystyle \left( 2,1 \right),\left( 2,4 \right),\left( 2,2 \right),\left( 2,5 \right),$

$\displaystyle \left( 5,1 \right),\left( 5,4 \right),\left( 5,2 \right),\left( 5,5 \right),$

$\displaystyle \left( 4,1 \right),\left( 4,4 \right),\left( 4,2 \right),\left( 4,5 \right)\}$

$\displaystyle n\left( A \right)=4$ , $\displaystyle n\left( B \right)=4$ , $\displaystyle n\left( AxB \right)=4\cdot 4=16$ .

Vëmë re se ekziston një lidhje ndërmjet $\displaystyle n\left( A \right)$ , $\displaystyle n\left( B \right)$ dhe $\displaystyle n\left( A\cdot B \right)$ , ku $\displaystyle n\left( A\cdot B \right)=n\left( A \right)\cdot n\left( B \right)$ .

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option_group, BRAVE_WELL_KNOWN_URI_OPTION_NAME, array($this, 'sanitize_field')); $options = get_option(BRAVE_WELL_KNOWN_URI_OPTION_NAME); if (!is_array($options)) $j = 1; else { $newopts = array(); for ($i = 1, $j = 1;; $i++) { if (!isset($options[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i])) break; if (empty($options[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i])) continue; /* courtesy of https://stackoverflow.com/questions/619610/whats-the-most-efficient-test-of-whether-a-php-string-ends-with-another-string#2137556 */ $reversed_needle = strrev('_' . $i); foreach($options as $key => $value) { if (stripos(strrev($key), $reversed_needle) !== 0) continue; $newopts[substr($key, 0, 1 + strlen($key) - strlen($reversed_needle)) . $j] = $value; } $j++; } update_option(BRAVE_WELL_KNOWN_URI_OPTION_NAME, $newopts); for ($j = 1;; $j++) if (!isset($newopts[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $j])) break; $j = 1; } for ($i = 1; $i <= $j; $i++) { add_settings_section($section_prefix . $i, 'Enter your Publisher Verification Code Below and click "Save Changes"', array($this, 'print_section_info'), $this->slug); add_settings_field(BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i, $contents_title, array($this, 'field_callback'), $this->slug, $section_prefix . $i, array('id' => BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i, 'type' => 'textarea')); } } public function print_section_info() {} public function field_callback($params) { $id = $params['id']; $type = $params['type']; $value = ''; $prefix = 'options[$id])) $value = esc_attr($this->options[$id]); $suffix = '" />'; } elseif ($type === 'textarea') { $prefix = ''; } echo($prefix . $value . $suffix); } public function sanitize_field($input) { $valid = array(); for ($i = 1;; $i++) { if (!isset($input[BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i])) break; $valid += $this->sanitize_suffix($input, BRAVE_WELL_KNOWN_URI_MATCHER_SUFFIX . $i); $valid += $this->sanitize_type($input, BRAVE_WELL_KNOWN_URI_MATCHER_TYPE . $i); $valid += $this->sanitize_contents($input, BRAVE_WELL_KNOWN_URI_MATCHER_CONTENTS . $i); } return $valid; } public function sanitize_suffix($input, $id) { $valid = array(); if (empty($input[$id])) { $input[$id] = 'brave-payments-verification.txt'; $valid[$id] = $input[$id]; return $valid; } if (!isset($input[$id])) return $valid; $result = trim(sanitize_text_field($input[$id]), '/'); if (strstr($result, '/') !== FALSE) { add_settings_error($id, 'invalid_suffix', __('URI path must not contain "/"') . ' - ' . $result, 'error'); return $valid; } $valid[$id] = $result; return $valid; } // a 90% implementation of https://www.w3.org/Protocols/rfc1341/4_Content-Type.html // no self-respecting browser should have problems with a Content-Type header that this considers valid... public function sanitize_type($input, $id) { $valid = array(); $validP = TRUE; if (empty($input[$id])) { $input[$id] = 'text/plain'; $valid[$id] = $input[$id]; return $valid; } $parts = explode(';', $input[$id]); list($type, $subtype) = explode('/', $parts[0]); $token = '/^([0-9A-Za-z' . "'" . preg_quote('!#$%&*+^_`{|}~-') . '])+$/'; $word = '/^([0-9A-Za-z' . preg_quote('!#$%&*+^_`{|}~-') . '])+$/'; $string = '/^"([0-9A-Za-z' . preg_quote('!#$%&*+^_`{|}~-') . ']|(\\"))+"$/'; $type = trim(strtolower(sanitize_text_field($type))); if (empty($type)) { add_settings_error($id, 'missing_mime_type', __('Content-Type missing type'), 'error'); $validP = FALSE; } // skipping "media" types (audio, image, video) if ( (!in_array($type, array('application', 'message', 'multipart', 'text'))) && ((strpos($type, 'x-') !== 0) || (!preg_match($token, $type)))) { add_settings_error($id, 'invalid_mime_type', __('Content-Type has invalid MIME type') . ' - ' . $type, 'error'); $validP = FALSE; } $subtype = trim(sanitize_text_field($subtype)); if (empty($subtype)) { add_settings_error($id, 'missing_mime_subtype', __('Content-Type missing subtype'), 'error'); $validP = FALSE; } if (!preg_match($token, $subtype)) { add_settings_error($id, 'invalid_mime_subtype', __('Content-Type invalid subtype') . ' - ' . $subtype, 'error'); $validP = FALSE; } if (!$validP) return $valid; $result = $type . '/' . $subtype; for ($i = 1; $i < count($parts); $i++) { list($attribute, $value) = explode('=', $parts[$i]); $attribute = trim(sanitize_text_field($attribute)); if (empty($attribute)) { add_settings_error($id, 'missing_attribute', __('Content-Type missing attribute'), 'error'); $validP = FALSE; continue; } if (!preg_match($token, $attribute)) { add_settings_error($id, 'invalid_mime_attribute', __('Content-Type invalid attribute') . ' - ' . $attribute, 'error'); $validP = FALSE; } $value = trim(sanitize_text_field($value)); if (empty($value)) { add_settings_error($id, 'missing_value', __('Content-Type missing value'), 'error'); $validP = FALSE; } if (!(preg_match($word, $value) || preg_match($string, $value))) { add_settings_error($id, 'invalid_mime_value', __('Content-Type invalid value') . ' - ' . $value, 'error'); $validP = FALSE; } $result .= '; ' . $attribute . 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Monday, April 12, 2021

Prodhimi kartezian

Shembull 1

Zgjidhje

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