{"id":3487,"date":"2023-05-24T20:15:49","date_gmt":"2023-05-24T20:15:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=3487"},"modified":"2025-08-29T21:02:33","modified_gmt":"2025-08-29T21:02:33","slug":"logic-basics-learn-it-5","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/logic-basics-learn-it-5\/","title":{"raw":"Logic Basics: Learn It 5","rendered":"Logic Basics: Learn It 5"},"content":{"raw":"

Equivalence<\/h2>\r\n

Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent<\/strong>, and that the converse and inverse are logically equivalent.<\/p>\r\n

\r\n[caption id=\"attachment_3490\" align=\"aligncenter\" width=\"741\"]\"A \u00a0Figure 1. The table shows how these variations are presented symbolically.[\/caption]\r\n<\/center>\r\n
\r\n
\r\n

equivalence<\/h3>\r\n

Statements [latex]p[\/latex] and [latex]q[\/latex] are said to be logically equivalent<\/strong> if they have the same truth value in every model.<\/p>\r\n

    \r\n\t
  • A conditional statement and its contrapositive are logically equivalent.<\/li>\r\n\t
  • The converse and inverse of a statement are logically equivalent.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n

    In other words, the original statement and the contrapositive must agree with each other; they must both be true, or they must both be false. Similarly, the converse and the inverse must agree with each other; they must both be true, or they must both be false.<\/p>\r\n

    DeMorgan's Laws<\/h2>\r\n

    There are two pairs of logically equivalent statements that come up\u00a0again and again in logic. They are prevalent\u00a0enough to be dignified by a special name: DeMorgan\u2019s laws.<\/b><\/p>\r\n

    Before we can dive into these laws, it is important to understand the term \"negation.\" Given a statement [latex]R[\/latex], the statement [latex]\\sim{R}[\/latex] is called the negation<\/b> of [latex]R[\/latex]. If [latex]R[\/latex]\u00a0is a complex statement, then it is often the case that its negation [latex]\\sim{R}[\/latex]\u00a0can\u00a0be written in a simpler or more useful form. The process of finding this\u00a0form is called negating<\/b> [latex]R[\/latex]. In proving theorems it is often necessary to\u00a0negate certain statements.<\/p>\r\n

    \r\n
    \r\n

    DeMorgan\u2019s laws<\/h3>\r\n
      \r\n\t
    1. The negation of a conjunction is equivalent to the disjunction of the negation of the statements making up the conjunction. To negate an \u201cand\u201d statement, negate each part and change the \u201cand\u201d to \u201cor\u201d.
      [latex]\\sim\\left(P{\\wedge}Q\\right)=({\\sim}P)\\vee\\left(\\sim{Q}\\right)[\/latex]<\/center><\/li>\r\n\t
    2. The negation of a disjunction is equivalent to the conjunction of the negation of the statements making up the disjunction. To negate an \u201cor\u201d statement, negate each part and change the \u201cor\u201d to \u201cand\u201d.
      [latex]\\sim\\left(P\\vee{Q}\\right)=\\left(\\sim{P}\\right)\\wedge\\left(\\sim{Q}\\right)[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section>\r\n
      The laws are named after Augustus De Morgan (1806\u20131871), who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians.\u00a0For example, in the 14th century, William of Ockham wrote down the words that would result by reading the laws out. Jean Buridan, in his Summulae de Dialectica<\/i>, also describes rules of conversion that follow the lines of De Morgan's laws.\u00a0
      \r\n
      \r\nStill, De Morgan is given credit for stating the laws in the terms of modern formal logic, and incorporating them into the language of logic. De Morgan's laws can be proved easily, and may even seem trivial.\u00a0Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.<\/section>\r\n

      DeMorgan\u2019s laws are actually very natural and intuitive. Consider the\u00a0statement [latex]\\sim\\left(P\\wedge{Q}\\right)[\/latex], which we can interpret as meaning that it is not the\u00a0case that both [latex]P[\/latex] and [latex]Q[\/latex] are true. If it is not the case that both [latex]P[\/latex] and [latex]Q[\/latex]\u00a0are true, then at least one of [latex]P[\/latex] or [latex]Q[\/latex] is false, in which case [latex]\\left(\\sim{P}\\right)\\vee\\left(\\sim{Q}\\right)[\/latex]\u00a0is\u00a0true. Thus [latex]\\sim\\left(P\\wedge{Q}\\right)[\/latex]\u00a0means the same thing as [latex]\\left(\\sim{P}\\right)\\vee\\left(\\sim{Q}\\right)[\/latex]. This law can also be verified using a truth table.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      [latex]P[\/latex]<\/b><\/th>\r\n[latex]Q[\/latex]<\/b><\/th>\r\n[latex]\\sim{P}[\/latex]<\/b><\/th>\r\n[latex]\\sim{Q}[\/latex]<\/b><\/th>\r\n[latex]P\\wedge{Q}[\/latex]<\/th>\r\n[latex]\\sim\\left(P\\wedge{Q}\\right)[\/latex]<\/strong><\/th>\r\n[latex]\\left(\\sim{P}\\right)\\vee\\left(\\sim{Q}\\right)[\/latex]<\/b><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      T<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\n<\/tr>\r\n
      T<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      F<\/td>\r\nT<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      F<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

       <\/p>\r\n

      DeMorgan\u2019s laws can be very useful. Suppose we happen to know that\u00a0some statement having form [latex]\\sim\\left(P\\vee{Q}\\right)[\/latex]\u00a0is true. The second of DeMorgan\u2019s\u00a0laws tells us that [latex]\\left(\\sim{Q}\\right)\\wedge\\left(\\sim{P}\\right)[\/latex]\u00a0is also true, hence [latex]\\sim{P}[\/latex] and [latex]\\sim{Q}[\/latex] are both true\u00a0as well. Being able to quickly obtain such additional pieces of information\u00a0can be extremely useful.<\/p>\r\n

      Here is a summary of some significant logical equivalences. Those that\u00a0are not immediately obvious can be verified with a truth table.<\/p>\r\n

      [latex]\\begin{array}{r@{\\hfill}l}
      \r\n\\text{Contrapositive law} & P\\rightarrow{Q}=(\\sim{Q})\\rightarrow(\\sim{P}) \\\\[6pt]
      \r\n\\text{DeMorgan's laws} & \\sim(P\\land{Q})=\\sim{P}\\lor\\sim{Q} \\\\
      \r\n& \\sim(P\\lor{Q})=\\sim{P}\\land\\sim{Q} \\\\[6pt]
      \r\n\\text{Commutative laws} & (P\\land{Q})={P}\\land{Q} \\\\
      \r\n& (P\\lor{Q})={P}\\lor{Q} \\\\[6pt]
      \r\n\\text{Distributive laws} & {P}\\land(Q\\lor{R})=({P}\\land{Q})\\lor(P\\land{R}) \\\\
      \r\n& P\\lor(Q\\land{R})=({P}\\lor{Q})\\land(P\\lor{R}) \\\\[6pt]
      \r\n\\text{Associative laws} & P\\land(Q\\land{R})=(P\\land{Q})\\land{R} \\\\
      \r\n& P\\lor(Q\\lor{R})=(P\\lor{Q})\\lor{R} \\\\
      \r\n\\end{array}[\/latex]<\/center>\r\n

       <\/p>\r\n

      Notice how the distributive law [latex]P\\wedge\\left(Q\\vee{R}\\right)=\\left(P\\wedge{Q}\\right)\\vee\\left(P\\wedge{Q}\\right)\\vee\\left(P\\wedge{R}\\right)[\/latex]\u00a0has the\u00a0same structure as the distributive law [latex]p\\left(q+r\\right)=p\\cdot{q}+p\\cdot{r}[\/latex]\u00a0from algebra.\u00a0Concerning the associative laws, the fact that [latex]P\\wedge\\left(Q\\wedge{R}\\right)=\\left(P\\wedge{Q}\\right)\\wedge{R}[\/latex]\u00a0means\u00a0that the position of the parentheses is irrelevant, and we can write this as [latex]P\\wedge{Q}\\wedge{R}[\/latex]\u00a0without ambiguity. Similarly, we may drop the parentheses in\u00a0an expression such as [latex]P\\vee\\left(Q\\vee{R}\\right)[\/latex].<\/p>\r\n

      But parentheses are essential when there is a mix of [latex]\\wedge[\/latex]\u00a0and [latex]\\vee[\/latex], as in [latex]P\\vee\\left(Q\\wedge{R}\\right)[\/latex]. Indeed, [latex]P\\vee\\left(Q\\wedge{R}\\right)[\/latex] and [latex]P\\vee\\left(Q\\wedge{R}\\right)[\/latex] and [latex]P\\vee\\left({Q}\\right)\\wedge{R}[\/latex] are not<\/b> logically equivalent.<\/p>\r\n

      [ohm2_question hide_question_numbers=1]7065[\/ohm2_question]<\/section>\r\n

      Here are some examples that illustrate how\u00a0DeMorgan\u2019s laws are used to negate statements involving \u201cand\u201d or \u201cor.\u201d<\/p>\r\n

      Consider negating the following statement.\r\n\r\n

      [latex]R[\/latex] : You can solve it by factoring or with the quadratic formula.<\/p>\r\n\r\n[reveal-answer q=\"102469\"]Show Solution[\/reveal-answer]
      \r\n[hidden-answer a=\"102469\"]Now, [latex]R[\/latex] means (You can solve it by factoring) [latex]\\vee[\/latex]\u00a0(You can solve it with the quadratic formula),\u00a0which we will denote as [latex]P\\vee{Q}[\/latex]. The negation of this is [latex]\\sim\\left(P\\vee{Q}\\right)=\\left(\\sim{P}\\right)\\wedge\\left(\\sim{Q}\\right)[\/latex].\r\n\r\n

      Therefore, in words, the negation of [latex]R[\/latex] is[latex]\\sim{R}[\/latex] : You can\u2019t solve it by factoring and you can\u2019t solve it with\u00a0the quadratic formula.<\/p>\r\n\r\nMaybe you can find [latex]\\sim{R}[\/latex]\u00a0without invoking DeMorgan\u2019s laws. That is good;\u00a0you have internalized DeMorgan\u2019s laws and are using them unconsciously.[\/hidden-answer]<\/section>\r\n

      Consider negating the following statement.\r\n\r\n

      [latex]R[\/latex] : The numbers [latex]x[\/latex] and [latex]y[\/latex] are both odd.<\/p>\r\n\r\n[reveal-answer q=\"441993\"]Show Solution[\/reveal-answer]
      \r\n[hidden-answer a=\"441993\"]This statement means [latex]\\left(x\\text{ is odd}\\right)\\wedge\\left(y\\text{ is odd}\\right)[\/latex], so its negation is:\r\n\r\n

      [latex]\\sim\\left[\\left(x\\text{ is odd}\\right)\\wedge\\left(y\\text{ is odd}\\right)\\right]=\\sim\\left(x\\text{ is odd}\\right)\\vee\\sim\\left(y\\text{ is odd}\\right)\\\\\\left(x\\text{ is odd}\\right)\\wedge\\left(y\\text{ is odd}\\right)=\\left(x\\text{ is even}\\right)\\vee\\left(y\\text{ is even}\\right)[\/latex]<\/p>\r\n

      Therefore the negation of [latex]R[\/latex] can be expressed in the following ways:<\/p>\r\n

      [latex]\\sim{R}[\/latex]: The number [latex]x[\/latex] is even or the number [latex]y[\/latex] is even.
      \r\n[latex]\\sim{R}[\/latex]: At least one of [latex]x[\/latex] and [latex]y[\/latex] is even.<\/p>\r\n

      [\/hidden-answer]<\/p>\r\n<\/section>\r\n

      [ohm2_question hide_question_numbers=1]7066[\/ohm2_question]<\/section>","rendered":"

      Equivalence<\/h2>\n

      Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent<\/strong>, and that the converse and inverse are logically equivalent.<\/p>\n

      \n
      \"A
      \u00a0Figure 1. The table shows how these variations are presented symbolically.<\/figcaption><\/figure>\n<\/div>\n
      \n
      \n

      equivalence<\/h3>\n

      Statements [latex]p[\/latex] and [latex]q[\/latex] are said to be logically equivalent<\/strong> if they have the same truth value in every model.<\/p>\n

        \n
      • A conditional statement and its contrapositive are logically equivalent.<\/li>\n
      • The converse and inverse of a statement are logically equivalent.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n

        In other words, the original statement and the contrapositive must agree with each other; they must both be true, or they must both be false. Similarly, the converse and the inverse must agree with each other; they must both be true, or they must both be false.<\/p>\n

        DeMorgan’s Laws<\/h2>\n

        There are two pairs of logically equivalent statements that come up\u00a0again and again in logic. They are prevalent\u00a0enough to be dignified by a special name: DeMorgan\u2019s laws.<\/b><\/p>\n

        Before we can dive into these laws, it is important to understand the term “negation.” Given a statement [latex]R[\/latex], the statement [latex]\\sim{R}[\/latex] is called the negation<\/b> of [latex]R[\/latex]. If [latex]R[\/latex]\u00a0is a complex statement, then it is often the case that its negation [latex]\\sim{R}[\/latex]\u00a0can\u00a0be written in a simpler or more useful form. The process of finding this\u00a0form is called negating<\/b> [latex]R[\/latex]. In proving theorems it is often necessary to\u00a0negate certain statements.<\/p>\n

        \n
        \n

        DeMorgan\u2019s laws<\/h3>\n
          \n
        1. The negation of a conjunction is equivalent to the disjunction of the negation of the statements making up the conjunction. To negate an \u201cand\u201d statement, negate each part and change the \u201cand\u201d to \u201cor\u201d.\n
          [latex]\\sim\\left(P{\\wedge}Q\\right)=({\\sim}P)\\vee\\left(\\sim{Q}\\right)[\/latex]<\/div>\n<\/li>\n
        2. The negation of a disjunction is equivalent to the conjunction of the negation of the statements making up the disjunction. To negate an \u201cor\u201d statement, negate each part and change the \u201cor\u201d to \u201cand\u201d.\n
          [latex]\\sim\\left(P\\vee{Q}\\right)=\\left(\\sim{P}\\right)\\wedge\\left(\\sim{Q}\\right)[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/section>\n
          The laws are named after Augustus De Morgan (1806\u20131871), who introduced a formal version of the laws to classical propositional logic. De Morgan’s formulation was influenced by algebraization of logic undertaken by George Boole, which later cemented De Morgan’s claim to the find. Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians.\u00a0For example, in the 14th century, William of Ockham wrote down the words that would result by reading the laws out. Jean Buridan, in his Summulae de Dialectica<\/i>, also describes rules of conversion that follow the lines of De Morgan’s laws.\u00a0<\/p>\n

          Still, De Morgan is given credit for stating the laws in the terms of modern formal logic, and incorporating them into the language of logic. De Morgan’s laws can be proved easily, and may even seem trivial.\u00a0Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.<\/section>\n

          DeMorgan\u2019s laws are actually very natural and intuitive. Consider the\u00a0statement [latex]\\sim\\left(P\\wedge{Q}\\right)[\/latex], which we can interpret as meaning that it is not the\u00a0case that both [latex]P[\/latex] and [latex]Q[\/latex] are true. If it is not the case that both [latex]P[\/latex] and [latex]Q[\/latex]\u00a0are true, then at least one of [latex]P[\/latex] or [latex]Q[\/latex] is false, in which case [latex]\\left(\\sim{P}\\right)\\vee\\left(\\sim{Q}\\right)[\/latex]\u00a0is\u00a0true. Thus [latex]\\sim\\left(P\\wedge{Q}\\right)[\/latex]\u00a0means the same thing as [latex]\\left(\\sim{P}\\right)\\vee\\left(\\sim{Q}\\right)[\/latex]. This law can also be verified using a truth table.<\/p>\n\n\n\n\n\n\n\n\n
          [latex]P[\/latex]<\/b><\/th>\n[latex]Q[\/latex]<\/b><\/th>\n[latex]\\sim{P}[\/latex]<\/b><\/th>\n[latex]\\sim{Q}[\/latex]<\/b><\/th>\n[latex]P\\wedge{Q}[\/latex]<\/th>\n[latex]\\sim\\left(P\\wedge{Q}\\right)[\/latex]<\/strong><\/th>\n[latex]\\left(\\sim{P}\\right)\\vee\\left(\\sim{Q}\\right)[\/latex]<\/b><\/th>\n<\/tr>\n<\/thead>\n
          T<\/td>\nT<\/td>\nF<\/td>\nF<\/td>\nT<\/td>\nF<\/td>\nF<\/td>\n<\/tr>\n
          T<\/td>\nF<\/td>\nF<\/td>\nT<\/td>\nF<\/td>\nT<\/td>\nT<\/td>\n<\/tr>\n
          F<\/td>\nT<\/td>\nT<\/td>\nF<\/td>\nF<\/td>\nT<\/td>\nT<\/td>\n<\/tr>\n
          F<\/td>\nF<\/td>\nT<\/td>\nT<\/td>\nF<\/td>\nT<\/td>\nT<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

           <\/p>\n

          DeMorgan\u2019s laws can be very useful. Suppose we happen to know that\u00a0some statement having form [latex]\\sim\\left(P\\vee{Q}\\right)[\/latex]\u00a0is true. The second of DeMorgan\u2019s\u00a0laws tells us that [latex]\\left(\\sim{Q}\\right)\\wedge\\left(\\sim{P}\\right)[\/latex]\u00a0is also true, hence [latex]\\sim{P}[\/latex] and [latex]\\sim{Q}[\/latex] are both true\u00a0as well. Being able to quickly obtain such additional pieces of information\u00a0can be extremely useful.<\/p>\n

          Here is a summary of some significant logical equivalences. Those that\u00a0are not immediately obvious can be verified with a truth table.<\/p>\n

          [latex]\\begin{array}{r@{\\hfill}l}
          \\text{Contrapositive law} & P\\rightarrow{Q}=(\\sim{Q})\\rightarrow(\\sim{P}) \\\\[6pt]
          \\text{DeMorgan's laws} & \\sim(P\\land{Q})=\\sim{P}\\lor\\sim{Q} \\\\
          & \\sim(P\\lor{Q})=\\sim{P}\\land\\sim{Q} \\\\[6pt]
          \\text{Commutative laws} & (P\\land{Q})={P}\\land{Q} \\\\
          & (P\\lor{Q})={P}\\lor{Q} \\\\[6pt]
          \\text{Distributive laws} & {P}\\land(Q\\lor{R})=({P}\\land{Q})\\lor(P\\land{R}) \\\\
          & P\\lor(Q\\land{R})=({P}\\lor{Q})\\land(P\\lor{R}) \\\\[6pt]
          \\text{Associative laws} & P\\land(Q\\land{R})=(P\\land{Q})\\land{R} \\\\
          & P\\lor(Q\\lor{R})=(P\\lor{Q})\\lor{R} \\\\
          \\end{array}[\/latex]<\/div>\n

           <\/p>\n

          Notice how the distributive law [latex]P\\wedge\\left(Q\\vee{R}\\right)=\\left(P\\wedge{Q}\\right)\\vee\\left(P\\wedge{Q}\\right)\\vee\\left(P\\wedge{R}\\right)[\/latex]\u00a0has the\u00a0same structure as the distributive law [latex]p\\left(q+r\\right)=p\\cdot{q}+p\\cdot{r}[\/latex]\u00a0from algebra.\u00a0Concerning the associative laws, the fact that [latex]P\\wedge\\left(Q\\wedge{R}\\right)=\\left(P\\wedge{Q}\\right)\\wedge{R}[\/latex]\u00a0means\u00a0that the position of the parentheses is irrelevant, and we can write this as [latex]P\\wedge{Q}\\wedge{R}[\/latex]\u00a0without ambiguity. Similarly, we may drop the parentheses in\u00a0an expression such as [latex]P\\vee\\left(Q\\vee{R}\\right)[\/latex].<\/p>\n

          But parentheses are essential when there is a mix of [latex]\\wedge[\/latex]\u00a0and [latex]\\vee[\/latex], as in [latex]P\\vee\\left(Q\\wedge{R}\\right)[\/latex]. Indeed, [latex]P\\vee\\left(Q\\wedge{R}\\right)[\/latex] and [latex]P\\vee\\left(Q\\wedge{R}\\right)[\/latex] and [latex]P\\vee\\left({Q}\\right)\\wedge{R}[\/latex] are not<\/b> logically equivalent.<\/p>\n