Pg. 367 conjunction: this and that = th disjunction: inclusive: this or that or both = this/that <<<< Math book usage exclusive: this or that, but not both = this (that) negation: this, but not that = this Pg. 373 implication: "p" (hypothesis) ">>" (implies) "q" (conclusion). conditional: "If p," (hypothesis) "then q" (conclusion). not biconditional: Just because "p >> q", doesn't necessarily mean its converse: "q >> p". examples: "All" prairie dogs are rodents. "If" an animal is a prairie dog", then" it is a rodent. "Every" prairie dog is a rodent. The fact that an animal is a prairie dog "implies" that it is a rodent. An animal is a prairie dog "only if" it is a rodent. Pg. 380 Rules of Logic valid argument: true premise; true premise; etc.; true conclusion. Direct Argument: If p is true, then q is true; p is true; Therefore(!), q is true. p >> q; p; !q. Indirect Argument: If p is true, then q is true; q is not true; Therefore, p is not true. p >> q; not q; !not p. Chain Rule: If p is true, then q is true; If q is true, then r is true; Therefore, if p is true, then r is true. p >> q; q >> r; !p >> r. Or Rule: p is true or q is true; p is not true; q is true. p or q; not p; !q. Pg. 381-2 Invalid Arguments converse error: p >> q; !q >> p. inverse error: p >> q; not p; !not q. Pg. 386 biconditional: p if and only if q; p = q. p >> q; q >> p; !p <> q. A biconditional does not imply a limitation; it could be part of a chain argument: p <> q; q <> r; p <> r. examples: conditionals: p is true; p >> truth. 1 is a real number; 1 >> real numbers. error: x >> y, without justification. reasoning: If 2 * 3 = 6, then 2 >> 6. If 2 * 6 not= 3, then 2 not>> 3. But if 2 * 3/2 = 3, then 2 >> 3. Therefore: x not>> y, automatically. inverse error: If 2 * 6 not= 3, then 2 not>> 3; not2; corrected to: not 2, but 1; !3. If 1 * 6 not= 3, then 1 not>> 3; !3. biconditionals: implications are conditional; implications <> conditional. + is positive; + <> positive. 2 * 3 = 6; 2 * 3 <> 6. 5/2 = 2.5; 5/2 <> 2.5. bicond. chain: 5/2 = 4/2 + 1/2 = 2 + 0.5 = 2.5 2.5 = 2 + 0.5 = 4/2 + 1/2 = 5/2 direct argument: 5/2 = 4/2 + 1/2; with its converse, 2.5 = 2 + 0.5 4/2 = 2; 2 = 4/2 1/2 = 0.5; 0.5 = 1/2 !2 + 0.5 = 2.5. !4/2 + 1/2 = 5/2 Pg. 388 definition: a biconditional; a conditional which excludes any other possibility in either direction of reasoning. example: An acute angle is between 0 and 90 degrees. poor definition, but a valid premise: An acute angle is 85 degrees. This does not exclude the possibility for an acute angle to be 65 degrees, etc. Pg. 395-7 two-column proof: itinerary: flow-chart: steps of progress linked by means (justifications) to get there. paragraph proof: same as flow-chart, but written out in plain english. Pg. 402 Algebra Postulates [1] Reflexive Property of Equality: A number is equal to itself. a = a [2] Substitution Property: If values are equal, one value may be substituted for the other. a = b [3] Distributive Property: A math operation between an expression and a set of of expressions; the individual expressions within the parenthetical set are seperated from each other by the same math operation; the primary math operation outside the set may be distributed individually among the elements within the set. If a, b, c, and d represent expressions, then: a * (b + c) = a * b + a * c While: a * (b + c * d) not= a * b + a * c * a * d [4] Addition Property of Equality: Addition is distributive over equality. a + (b = c) >>> a + b = a + c (b = c) + a >>> b + a = c + a [5] Subtraction Property of Equality: Subtraction is distributive over equality. a - (b = c) >>> a - b = a - c; a + -1 * (b = c) >> From [4] above. (b = c) - a >>> a - b = a - c; -a + (b = c) >> From [4] above. [6] Multiplication Property of Equality: Multiplication is distributive over equality. a * (b = c) >>> a * b = a * c (b = c) * a >>> b * a = c * a [7] Division Property of Equality: Division is distributive over equality. a / (b = c) >>> a / b = a / c; (*) a * (b = c)^-1 >>> a * (b^-1 = c^-1) [3] >>> a * b^-1 = a * c^-1 [3] (b = c) / a >>> b / a = c / a; (b = c) * a^-1 >>> b * a^-1 = c * a^-1 [3] Both use [3]. Exponentiation is distributive over equality if it is considered to be the same as multiplication. But it has to be used twice at the (*) starred line, because of exponentiation's conventional precedence over multiplication. Pg. 27, Selected Answers Division by Zero The world of magnitude is a circle, not a flat line segment. All systems of enumeration are relative and have a finite capacity for fullness and emptiness. Wherever a system partakes of an upper or a lower limit, there it touches absolute magnitude----the indefinable. But magnitude is a ring, so there are two versions of relative absolute: infinity and zero. Both are versions of the same thing seen from two different perspectives. Imagine sitting on a point marked on a circle; this is its reference. The circle is marked with a fixed number of magnitude divisions; this is its capacity for relativity. No markings would constitute an absolute frame of reference. The divisions run positively and in increasing positive integer value in one direction; negative with increasing negative integer value in the opposite direction. Although at any one time it is either one version of a ring or its compliment, potentially it is a double ring linked at every marking including the reference. First, choose a positive view: the reference will be named zero. Any motion away from the reference will be by adding positive integers to zero. Eventually, the capacity of the system will be reached: infinity, which had been the starting point. Second, take the reverse condition, negative: the reference will be infinity. Any motion away from the starting point will be making an ever widening hole in infinity by subtracting an ever increasing something from the upper limit. Eventually, zero will be reached (originally infinity). Notice there are two ways of changing a limit into its opposite: by counting from one extreme to the other or by multiplying the reference by a -1 when standing on the reference and merely inverting direction. If the circle has only ten markings on it, one of them the reference, then: positive >>>> +1 >>> +0, +1, +2, +3, +4, +5, +6, +7, +8, +9 direction -9, -8, -7, -6, -5, -4, -3, -2, -1, -0 <<< -1 <<< negative direction Notice that the single digit values are paired in each column with a compliment. In computer architecture, this is called "one's compliment". For the purposes of subtraction, a negative number is inverted into its one's compliment, then a special form of wrap-around carry-over addition is performed reflecting the structure of a ring. Inversion is the equivalent of multiplying by a -1. Working with a three digit capacity and single digit examples, the left two digit positions filled in with either 0's or 9's act as sign digits; 0 means positive and 9 means negative. Usually, the left-most digit is reserved for signing by keeping it off-limits to addition or subtraction, but allowing it to invert into its opposite upon multiplication by a -1. Using the above two strings of digits as a table of compliments: 111 >> carry-over digits; since the +8 = 008 = 008 left-most carry-over 1 has -4 = 004 * -1 = +995 nothing to add to, it is --- --- carried "around" to the +4 003 right-most digit and added +001 << here; (this wrap-around can --- occur as many as three times 004 until the dust settles); --- the result is interpreted as +4 >> being a positive four. *** >> nothing to carry-over; +3 = 003 = 003 -7 = 007 * -1 = 992 --- --- -4 995 * -1 = -004 = -4 >> Voila! (french: behold) *** >> nothing to carry-over; +1 = 001 = 001 -1 = 001 * -1 = 998 --- --- 0 999 * -1 = -000 = -0 (Of course in computers everything is done in base two, but one's compliment will work with any base so long as the math system has an even number of different possible signs.) Mathematicians in the early days of computering unfortunately freaked-out over the occurrence of negative zero. They decided to conform to the standards of fear and ignorance that is strictly enforced to this day by performing what is called "two's compliment" arithmetic within computer operations---thus, saving face. Computers never have this little problem now, but they fail to teach us anything about the universe either. Calculators never have this problem, they do things differently. So, in a one-digit universe: 0 = lower limit; 9 = upper limit; -0 = upper limit; -9 = lower limit 0^1 = lower limit; 9^1 = upper limit; 0^-1 = upper limit; 9^-1 = lower limit Although multiplying any number inverts it into its one's compliment, raising a number to a negative power doesn't automatically invert it. It has to be a limit in order to invert: 2^-1 not= 7^1, but 2^-1 = 1/2 Also, 2^-1 = 002^998 >>> this is useful for doing logarithms, but it still has to be interpreted correctly at the end of operations. Raising a limit to a negative power doesn't have an analogue in computerese, but is illustrated subjectively by an example: x * y = 1 Solving for y yields: y = 1/x Using x as the input value from 1 to infinity yields a range of y from 1 to zero. At least this helps to define division by infinity. But what about division by zero? Looking at the graph of this function begs the question: what do we have against dividing by zero, especially since the graph points out the trend to where the answer is headed; namely, infinity. Of course we could have substituted a negative infinity in place of either of the two zero's to postu- late a situation and attempt a solution in a different manner to see what happens: Using: 9 = infinity; 0 = zero; And: -0 = infinity; -9 = zero; y = 1/x 0 = 1/9 9 * 0 =? 1 [%] 9 * 0 = 0 9 * -9 = ? (8 >> carry-over wraped-around to 9 * 9 * -1 = 9 *9 --- 1 +8 >> here and added --- [*] 9 << gives, * -1 >> but when the multiplication ----- is completed the same result 0 >> occurs; no matter what we divide by, zero or infinity, the constant gets squashed to zero when the function is put back into the form of multiplication: infinity = (1: old constant) / zero Or: zero = (1: old constant) / infinity zero * infinity = (zero: new constant) >> line [%]; What in effect has happened when we divide by a limit is the function flattens into an asymptotic function We already know: zero * zero = zero And: infinity * infinity = infinity >> line [*], but these examples don't help with the problem of defining division by zero. Of course, we are allowed to take a step back and see how it looks: Using: & = infinity; 0 = zero; y = 1/x Yields: 0 = 1/&, as x approaches infinity; & * 0 = 0 But wait; we weren't allowed to do the inverse: y = 1/x Yields: & = 1/0, as x approaches zero; 0 * & = 0 Should there be any difference between: 0 * & = & * 0 = 0? Is there any problem with solving for a function alternately, first for x and then for y, and then making a composite graph of the two solution systems? We do that all the time by setting the function equal to zero and alternately solving for x and y: x * y = 0 y = 0/x Yields: 0 = 0/x, no matter what x is; And: 0 = 0/y, no matter what y is. This points to only one conclusion: That infinity and zero are limits to magnitude and are compliments to each other. This thinking isn't new. In the I Ching it states: whenever anything goes to far in the extreme, it changes into its opposite (a closed, looped universe; the dragon biting its tail image). The capacity of a system to embody relatively absolute limits is suggested by an idea stated as the topic of Maharishi Mahesh Yogi's 18th(?) course on The Science of Creative Intelligence: "Fullness of fullness; fullness of emptiness".