Necessary and/or Sufficient Conditions with Modular MathDate: 12/01/2006 at 19:24:49 From: Sam Subject: Modulo If x is congruent to y modulo 10 whenever there exists an integer k such that x - y = 10k, identify which of the conditions below are "sufficient", "necessary", "necessary and sufficient" or none of these in order that x is congruent to y modulo 10. (a) x - y = 50 (b) x - y = 5 (c) x - y is divisible by 10 (d) x - y is divisible by 5 (e) x - y is divisible by both 5 and 2 (f) x and y are both even I don't understand what is meant by sufficient, necessary, and sufficient and necessary. Date: 12/01/2006 at 23:05:30 From: Doctor Peterson Subject: Re: Modulo Hi, Sam. A condition A is "necessary" for a result B if B is true ONLY if A is true; that is, A HAS TO be true in order for B to be true. We say that B implies A, or "B only if A". A condition A is "sufficient" for a result B if B is true WHENEVER A is true; that is, A is ENOUGH to force B to be true. We say that A implies B, or "B if A". A condition A is "necessary AND sufficient" for a result B if both of the above are valid; we say that A is true IF AND ONLY IF B is true. A and B are equivalent. As an example, suppose I asked whether "x and y are both even" is a necessary and/or sufficient condition for the product xy to be even. Since IF x and y are both even, THEN xy is even, it is a sufficient condition; knowing they are both even is enough to be certain that the product is even. But xy will also be even if only one of the factors is even; so having both even is NOT necessary. See also: Necessary and/or Sufficient http://mathforum.org/library/drmath/view/60701.html Now look at your definition: x is congruent to y modulo 10 whenever there exists an integer k such that x - y = 10k. In ordinary language, x and y are congruent (mod 10) if they differ by a multiple of 10. So we can take the first condition: x - y = 50 This says that x and y differ by 50, which is a multiple of 10; so they WILL be congruent. This is a sufficient condition. But is it necessary? What if x and y differed by, say, 40? They would still be congruent, but this condition would not be true. So it is NOT necessary. If you have any further questions, feel free to write back. Give me whatever answers you have, and we can discuss any mistakes you've made. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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