L07_IntroLogic

# L07_IntroLogic - 1-1COMP170Discrete Mathematical Toolsfor...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 3.1, pp. 91-101Intro to LogicVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen2-13.1 Equivalence and Implication•Equivalence of Statements•Truth Tables•DeMorgan’s Laws•Implication•If and Only If3-1Equivalence of Statements3-2Equivalence of Statements(1) if ((i+j≤p+q) && (i≤p) &&((j > q) || (List1[i]≤List2[j])))(2)List3[k] = List1[i](3)i = i+1(4) else(5)List3[k] = List2[j](6)j = j+1(7) k = k+1(1) if (((i+j≤p+q) && (i≤p) && (j > q))|| ((i+j≤p+q) && (i≤p)&& (List1[i]≤List2[j])))(2)List3[k] = List1[i](3)i = i+1(4) else(5)List3[k] = List2[j](6)j = j+1(7) k = k+13-3Equivalence of Statements(1) if ((i+j≤p+q) && (i≤p) &&((j > q) || (List1[i]≤List2[j])))(2)List3[k] = List1[i](3)i = i+1(4) else(5)List3[k] = List2[j](6)j = j+1(7) k = k+1(1) if (((i+j≤p+q) && (i≤p) && (j > q))|| ((i+j≤p+q) && (i≤p)&& (List1[i]≤List2[j])))(2)List3[k] = List1[i](3)i = i+1(4) else(5)List3[k] = List2[j](6)j = j+1(7) k = k+1Consider the two piecesof code on the left.They are taken from twodifferentversionsofMergesort. Do they dothe same thing?&& =“and”||=“or”3-4Equivalence of Statements(1) if ((i+j≤p+q) && (i≤p) &&((j > q) || (List1[i]≤List2[j])))(2)List3[k] = List1[i](3)i = i+1(4) else(5)List3[k] = List2[j](6)j = j+1(7) k = k+1(1) if (((i+j≤p+q) && (i≤p) && (j > q))|| ((i+j≤p+q) && (i≤p)&& (List1[i]≤List2[j])))(2)List3[k] = List1[i](3)i = i+1(4) else(5)List3[k] = List2[j](6)j = j+1(7) k = k+1Consider the two piecesof code on the left.They are taken from twodifferentversionsofMergesort. Do they dothe same thing?&& =“and”||=“or”Code is sameexcept for line 14-1(1) ((i+j≤p+q) && (i≤p) &&((j > q) || (List1[i]≤List2[j])))(1’) (((i+j≤p+q) && (i≤p) && (j > q))|| ((i+j≤p+q) && (i≤p)&& (List1[i]≤List2[j])))&&=“and”||=“or”4-2(1) ((i+j≤p+q) && (i≤p) &&((j > q) || (List1[i]≤List2[j])))(1’) (((i+j≤p+q) && (i≤p) && (j > q))|| ((i+j≤p+q) && (i≤p)&& (List1[i]≤List2[j])))&&=“and”||=“or”Are they equivalent?4-3(1) ((i+j≤p+q) && (i≤p) &&((j > q) || (List1[i]≤List2[j])))(1’) (((i+j≤p+q) && (i≤p) && (j > q))|| ((i+j≤p+q) && (i≤p)&& (List1[i]≤List2[j])))&&=“and”||=“or”Are they equivalent?Let’s rewrite usings∼(i+j≤p+q)t∼(i≤p)u∼(j>q)v∼(List[i]≤List2[j])4-4(1) ((i+j≤p+q) && (i≤p) &&((j > q) || (List1[i]≤List2[j])))(1’) (((i+j≤p+q) && (i≤p) && (j > q))|| ((i+j≤p+q) && (i≤p)&& (List1[i]≤List2[j])))&&=“and”||=“or”Are they equivalent?Let’s rewrite usings∼(i+j≤p+q)t∼(i≤p)u∼(j>q)v∼(List[i]≤List2[j])(1)sandtand (uorv)(1’) (sandtandu) or (sandtandv)4-5(1) ((i+j≤p+q) && (i≤p) &&((j > q) || (List1[i]≤List2[j])))(1’) (((i+j≤p+q) && (i≤p) && (j > q))|| ((i+j≤p+q) && (i≤p)&& (List1[i]≤List2[j])))&&=“and”||=“or”Are they equivalent?Are they equivalent?...
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L07_IntroLogic - 1-1COMP170Discrete Mathematical Toolsfor...

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