{"id":2181,"date":"2023-10-02T10:34:30","date_gmt":"2023-10-02T10:34:30","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=2181"},"modified":"2023-10-02T10:35:05","modified_gmt":"2023-10-02T10:35:05","slug":"week-10-ss1-second-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-10-ss1-second-term-further-mathematics-notes\/","title":{"rendered":"Week 10 &#8211; SS1 Second Term Further Mathematics Notes"},"content":{"rendered":"<p>\u00a0<\/p>\n<h3>WEEK  TEN \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0DATE\u2026\u2026\u2026\u2026\u2026<br \/>\n<\/h3>\n<p><strong>TOPIC:<\/strong> Logical reasing continues <strong>CONTENT: <\/strong><\/p>\n<ul>\n<li>Connectives; (Disjunction and conjunction)\n<\/li>\n<li>Tautology and contradiction<strong><br \/>\n\t\t\t\t<\/strong>\n\t\t<\/li>\n<\/ul>\n<p><strong>Disjunction: <\/strong>In disjunction two statement can be combined by the use of the connective to <strong>the truth table. <\/strong>The truth table technique is used to establish whether or not two logical statement are equivalent. Let p = He is a pastor and q = He is a singer <\/p>\n<p>\u00a0The above statement can be written as either he is a pastor or he is a singer.<br \/>\nHence, in logical symbols; the statement can be written as p or q, where or means v i.epvq.<br \/>\n<strong>NOTE: <\/strong> the statement Pvq is false when both p and q are the false otherwise pvq is true. The truth table for the above statement is given or presented as: <\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>p <\/td>\n<td>q <\/td>\n<td>Pvq <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>T <\/td>\n<td>T <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>F <\/td>\n<td>T <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>T <\/td>\n<td>T <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<strong>CONJUNCTION: <\/strong>When the connective and is used to combine two statement thus, we have <strong>conjunction. <\/strong>Let p = Lagos is in Nigeria<br \/>\nLet q = 3 is an odd number Thus, the above statement can be combined using the connective &#8220;and&#8221; as in : Lagos is in Nigeria and 3 is an odd number and it can be written as; p and q, where and is symbolically represented as <em>\u02c4<\/em> i.e <em>\u02c4<\/em> means &#8220;and&#8221;. Hence, p and q = p<em>\u02c4<\/em>q.<br \/>\nThe above statement can be illustrated using a truth table.<br \/>\n<strong>NOTE: <\/strong>the statement p<em>\u02c4<\/em>q is true when the sub statement p and q are both true otherwise p<em>\u02c4<\/em>q is False. <strong><br \/>\n\t\t\t<\/strong><\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>p <\/td>\n<td>q <\/td>\n<td>p<em>\u02c4<\/em>q <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>T <\/td>\n<td>T <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>T <\/td>\n<td>F <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<strong>TAUTOLOGY: <\/strong>A compound statement which is always true irrespective of the truth values of the sub statement is called <strong>TUATOLOGY<\/strong>. It is represented as T. <\/p>\n<p>\u00a0<strong>Example<\/strong>: Use the truth table to show that the statement pv~p is a tautology.  <\/p>\n<p>\u00a0<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>p <\/td>\n<td>~p <\/td>\n<td>pv~p <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>F <\/td>\n<td>T <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>F <\/td>\n<td>T <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>T <\/td>\n<td>T <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>T <\/td>\n<td>T <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p> From the above table it can be observed that the last column has the truth value T. Hence, the statement is<br \/>\n<strong>TAUTOLGY. <\/strong><\/p>\n<p>\u00a0<strong>CONTRADICTION: <\/strong>A compound statement which is always False irrespective of the truth value of the sub statement is called <strong>CONTRADICTION. <\/strong> It is usually denoted by F. <\/p>\n<p>\u00a0<strong>Example<\/strong>: Use the truth table to show that the statement p<em>\u02c4<\/em>~p is a tautology.  <\/p>\n<p>\u00a0<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>p <\/td>\n<td>~p <\/td>\n<td>p<em>\u02c4<\/em>~p <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>T <\/td>\n<td>F <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>T <\/td>\n<td>F <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>From the above table it can be observed that the last column has the truth value F. Hence, the statement is<br \/>\n<strong>CONTRADICTION.<\/strong><\/p>\n<p>\u00a0<strong>EVALUATION: <\/strong><br \/>\n\t    1.Copy and complete the truth table below: <\/p>\n<p>\u00a0<\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td>P <\/td>\n<td>q <\/td>\n<td>r <\/td>\n<td>qvr <\/td>\n<td>~p<em>\u02c4<\/em>(qvr) <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>T <\/td>\n<td>T <\/td>\n<td> \u00a0<\/td>\n<td>T <\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>T <\/td>\n<td>F <\/td>\n<td> \u00a0<\/td>\n<td> \u00a0<\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>F <\/td>\n<td>T <\/td>\n<td>T <\/td>\n<td> \u00a0<\/td>\n<\/tr>\n<tr>\n<td>T <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>T <\/td>\n<td>T <\/td>\n<td> \u00a0<\/td>\n<td> \u00a0<\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>T <\/td>\n<td>F <\/td>\n<td> \u00a0<\/td>\n<td> \u00a0<\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>F <\/td>\n<td>T <\/td>\n<td> \u00a0<\/td>\n<td> \u00a0<\/td>\n<\/tr>\n<tr>\n<td>F <\/td>\n<td>F <\/td>\n<td>F <\/td>\n<td> \u00a0<\/td>\n<td>F <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a02. Use the truth table technique to establish the following results:<br \/>\n(i) \u00a0\u00a0\u00a0\u00a0p<em>\u02c4<\/em>q = q<em>\u02c4<\/em>p<br \/>\n( ii.) pv(q<em>\u02c4<\/em>r) = (pvq)vr<br \/>\n(iii) {p<em>\u02c4<\/em>(~pvq)}Vq is a tautology <\/p>\n<p>\u00a0<strong>GENERAL EVALUATION: <\/strong><\/p>\n<ol>\n<li>Draw the truth table for ~ (p\u2192 ~q) Using the truth tables, prove that:\n<\/li>\n<li>p<br \/>\n\t\t\t<em>\u02c4<\/em>{(~p<em>\u02c4<\/em>p)V(~p~<em>\u02c4<\/em>q)} is a contradiction.\n<\/li>\n<li>{(pv~q)<br \/>\n\t\t\t<em>\u02c4<\/em>(~pv~q)}Vq is tautology.\n<\/li>\n<\/ol>\n<p>\u00a0<strong>Reading Assignment: F\/Maths Project 2 pages 30 Exercise 3 Q 9 and 12 <\/strong><\/p>\n<p>\u00a0<\/p>\n<h3>WEEKEND ASSIGNMENT<br \/>\n<\/h3>\n<p>1.  Let p = She is short and q = She is beautiful. Write each of the following in symbolic form using p and q.<br \/>\n(i) She is short and beautiful (ii) She is short and but not beautiful (iii) It is false that she is tall and beautiful (iv) She is neither short nor beautiful.<br \/>\nUse the truth table technique to show that <\/p>\n<ol>\n<li>p\u2194q = (p\u2192q)<br \/>\n\t\t\t<em>\u02c4<\/em>(q\u2192p)\n<\/li>\n<li>(p<em>\u02c4<\/em>q)<em>\u02c4<\/em>~(pvq) is a contradiction.\n<\/li>\n<li>(~pv~q)v~ (pvr)v(qvr) is a tautology.\n<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0 WEEK TEN \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0DATE\u2026\u2026\u2026\u2026\u2026 TOPIC: Logical&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,188],"tags":[],"class_list":["post-2181","post","type-post","status-publish","format-standard","hentry","category-posts","category-second-term-ss1-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2181","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=2181"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2181\/revisions"}],"predecessor-version":[{"id":2182,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2181\/revisions\/2182"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=2181"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=2181"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=2181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}