{"id":3295,"date":"2023-10-04T14:00:39","date_gmt":"2023-10-04T14:00:39","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=3295"},"modified":"2023-10-04T14:01:39","modified_gmt":"2023-10-04T14:01:39","slug":"week-7-8-and-9-ss2-second-term-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-7-8-and-9-ss2-second-term-mathematics-notes\/","title":{"rendered":"Week 7, 8 and 9 – SS2 Second Term Mathematics Notes"},"content":{"rendered":"

WEEK SEVEN
\n<\/strong> REVISION AND MID TERM EXAMINATION
\n<\/strong> WEEK 8
\n<\/strong>
\n\u00a0OPERATIONS IN ALGEBRAIC FRACTIONS<\/p>\n

    \n
  1. \n
    Simplify\n<\/div>\n

    \u00a0= <\/p>\n

    \u00a0<\/p>\n

    \u00a0<\/p>\n

    \u00a0<\/li>\n

  2. \n
    Simplify\n<\/div>\n

    = <\/p>\n

    \u00a0<\/li>\n

  3. \n
    \n\t\t\t\t<\/div>\n

    \u00a0Factorize each term to obtain
    \n
    \n\t\t\t\t\t<\/em>
    \n\u00a0Change to x and then invert to obtain<\/p>\n<\/li>\n<\/ol>\n

    \u00a0But x \u2013 y = -1(y \u2013 x)
    \n = – (2x + y) = -2x \u2013 y<\/p>\n

    \u00a0
    \n\u00a0
    \n\u00a0WRAP UP AND ASSESSMENT
    \n<\/strong>You can simplify fractions by adding, subtracting, multiplying or dividing them. To simplify a fraction means to reduce it to its lowest term. To do this, factorize both the numerator and denominator fully.
    \nThen cancel the common factors.<\/p>\n

    \u00a0Simplify
    \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \nIi. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0iii.\u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0
    \n\u00a0\"\"\/iv.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0v.\u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0
    \n\u00a0TICKET OUT
    \n<\/strong>i.\u00a0\u00a0\u00a0\u00a0if = x, evaluate <\/p>\n

    \u00a0ii. \u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0iii.\u00a0\u00a0\u00a0\u00a0if x:y = 12:5 evaluate <\/p>\n

    \u00a0
    \n\u00a0WEEK 9
    \n<\/strong>LOGIC
    \n<\/strong>A Proposition is a statement or sentence that either true or false but not both. A simple statement or proposition is a statement containing no connectives. In other words a proposition is considered simple. If it cannot be broken up into sub-propositions.
    \nOn the other hand, a compound proposition is made up of two or more propositions joined by the connectives. These connectives are and, or, if \u2026.. Then, if and only if. They are also called logic operators.
    \nLogic operator symbol
    \n<\/strong>And\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0^\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n\t\t\tOr\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u02c5
    \n\"\"\/If\u2026..then\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \nIf and only if\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u21d4
    \nnot\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \nIF P AND Q ARE TWO STATEMENTS (OR PROPOTIONS) THEN {CONDITIONAL STATEMENTS AND INDIRECT PROOFS}<\/p>\n

      \n
    1. The statement p ^ q is called the conjunction of p ^ q. This, p ^ q means p and q.\n<\/li>\n
    2. The statement p v is called the disjunction of p and q. This, p v q means either p or q or both p and q.\n<\/li>\n
    3. \"\"\/\"\"\/The statement p is called the conditional of p and q. a conditional is also known as implication p q means if p then q or p implies q.\n<\/li>\n
    4. \"\"\/\"\"\/The converse of the conditional statement if p then q is the conditional statement if q then p, (ie) the converse of p q is q p.\n<\/li>\n
    5. \"\"\/\"\"\/The inverse of the conditional statement if p then q is the conditional statement if not p then not q. i.e. The inverse of p q is
      \n\t\t\t\t<\/strong>p q<\/strong>\n\t\t\t<\/li>\n
    6. \"\"\/\"\"\/The contra positive of the conditional statement if p then q is the conditional statement if not q then not p. i.e the contra positive of p q is q <\/strong> p.\n<\/li>\n
    7. \"\"\/\"\"\/\"\"\/The statement p \u21d4 q is called the bi conditional of p and q, where the symbol \u21d4 means if and only if (or if for short). This, p\u21d4q means p q means p q and q p\n<\/li>\n
    8. The statement
      \n\t\t\t\t<\/strong>p is known as the negation of p Thus
      \n\t\t\t\t<\/strong>p means not p or “it is false that p \u2026\u2026’ or “it is not true that p\u2026’\n<\/li>\n
    9. When a compound proposition is always true for every combination of values of its constituent statements. It is called a TAUTOLOGY<\/strong>. On the other hand, when the compound proposition is always false it is called a CONTRADICTION<\/strong>.\n<\/li>\n<\/ol>\n

      \u00a0THE TRUTH TABLES
      \n<\/strong>The Truth or falsify of a proposition is its truth values. A proposition that is true has a truth value T and a proposition that is false has a truth value of F.<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      CONJUCTION<\/strong><\/td>\nDISJUNCTION<\/strong><\/td>\nCONDITIONAL <\/strong><\/td>\n<\/tr>\n
      P q p ^ q<\/td>\nP q p v q<\/td>\n\"\"\/P q p q<\/td>\n<\/tr>\n
      T T T<\/td>\nT T T<\/td>\nT T T<\/td>\n<\/tr>\n
      T F F<\/td>\nT F T<\/td>\nT F F<\/td>\n<\/tr>\n
      F T F<\/td>\nF T T<\/td>\nF T T <\/td>\n<\/tr>\n
      F F F<\/td>\nF F F<\/td>\n F F T<\/td>\n<\/tr>\n
      P ^ Q is true when both p and q are true <\/td>\nP v q is false when both p and q are false<\/td>\n\"\"\/P is false when p is T & q is F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

      \u00a0<\/p>\n

      \n\n\n\n\n\n\n\n\n
      BICONDITIONAL<\/strong><\/td>\nNEGATION<\/strong><\/td>\n<\/tr>\n
      P q p \u21d4 q<\/td>\nP P<\/strong><\/td>\n<\/tr>\n
      T T T<\/td>\nT F<\/td>\n<\/tr>\n
      F T F<\/td>\nF T<\/td>\n<\/tr>\n
      F F T<\/td>\n\"\"\/\"\"\/Recall that other symbols used instead of are pI<\/sup> or <\/strong>p p<\/td>\n<\/tr>\n
      P \u21d4 q is true when both p and q are either both true and both false.<\/td>\n\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

      \u00a0
      \n\u00a0
      \n\u00a0ASSOCIATED TERMS IN ALGEBRA OF SETS AND ALGEBRA OF PROPOSITIONS
      \n<\/strong>The structure of algebra of sets and the algebra of propositions look the same. The associated term are given in the table below.<\/p>\n

      \n\n\n\n\n\n\n\n\n\n\n\n
      Algebra of sets<\/td>\nAlgebra of Proposition<\/td>\n<\/tr>\n
      Sets A, B, C<\/td>\nPropositions p, q, r,<\/td>\n<\/tr>\n
      Union U<\/td>\nDisjunction V<\/td>\n<\/tr>\n
      Intersection <\/td>\nConjunction ^<\/td>\n<\/tr>\n
      Complement A’<\/td>\nNegation P<\/td>\n<\/tr>\n
      Universal set <\/td>\nTautology, t<\/td>\n<\/tr>\n
      Nill (empty) set <\/td>\nSelf-contradiction f<\/td>\n<\/tr>\n
      Is a subject of C<\/td>\n\"\"\/Implies <\/td>\n<\/tr>\n
      Equals =<\/td>\nIs equivalent to <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

      \u00a0
      \n\u00a0\"\"\/\"\"\/\"\"\/\"\"\/\"\"\/For example in,
      \n\"\"\/
      \n\t\t\"\"\/<\/p>\n

      \u00a0\"\"\/<\/p>\n

      \u00a0
      \n\u00a0\"\"\/A C B\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\u00a0p q means p implies q
      \nMeans A is a proper subset of B<\/p>\n

      \u00a0THE VALIDITY OF AN ARGUMENT
      \n<\/strong>There are two forms of reasoning used in mathematics namely, inductive reasoning and deductive reasoning.
      \nInductive reasoning usually lacks generality because not all possibilities have been exhausted, when we use inductive reasoning, we base our conclusions on observation or experiences.<\/p>\n

      \u00a0On the other hand, deductive reasoning is the process of showing that certain statements are accepted as true. In deductive reasoning all possibilities have been exhausted and therefore a generalized conclusion can be made.<\/p>\n

      \u00a0Valid argument may be referred to a deductive arguments because deductive reasoning is based on conclusions reached from valid arguments. In deductive reasoning, we start with assumptions (also called hypotheses or premises) and then draws a conclusion based on those assumptions.<\/p>\n

      \u00a0An argument may be described as a set of statements or proposition called the premises which leads to a conclusion. Let P1<\/sub>, P2<\/sub>, P3<\/sub> \u2026\u2026\u2026..Pn<\/sub> represent the premises of an argument and C represents the conclusion. A valid argument is one in which if the premises P1<\/sub>, P2<\/sub>, P3<\/sub>\u2026\u2026 Pn<\/sub> are all true, the conclusion C will always be true. In other words, an argument is said to be valid if the conjunction of the compound statement i.e P1<\/sub> ^ P2<\/sub> ^ P3<\/sub>\u2026\u2026 ^ Pn<\/sub> is tautology. If an argument is not valid, it is called invalid or a fallacy. This, argument is valid if the conclusion follows from the hypotheses.
      \nWRAP UP AND ASSESSMENT
      \n<\/strong>A Proposition is a statement that is either true (T) or False (F) but not both.
      \nA compound statement or proposition is made up of two or more simple statements joined by the connectives. Ex 10.1 No 1, 2
      \nTICKET OUT
      \n<\/strong>Ex 10.1 No 3, 4<\/p>\n","protected":false},"excerpt":{"rendered":"

      WEEK SEVEN REVISION AND MID TERM EXAMINATION WEEK 8 \u00a0OPERATIONS IN ALGEBRAIC FRACTIONS Simplify \u00a0=…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,264],"tags":[],"class_list":["post-3295","post","type-post","status-publish","format-standard","hentry","category-posts","category-second-term-ss2-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3295","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=3295"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3295\/revisions"}],"predecessor-version":[{"id":3296,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3295\/revisions\/3296"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=3295"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=3295"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=3295"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}