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Scalars and Vectors We write r=<a, b> where a and b are the components of vector v Note: Both and r represent vectors, and will be used interchangeably Scalars and Vectors (2) The components a and b are both scalar quantities The position vector, or directed line segment from the origin to point P(a,b), is r=<a, b>
Slide 1 R = (A + B) + C = A + (B + C) Associative Property of Vectors A B O Q P A + B C R= A+B+C R B + C Home Next Previous If the two vectors (to be added) can be represented both in magnitude and direction by the two adjacent sides of a parallelogram, then the resultant is represented completely (both in magnitude and direction) by the diagonal of
CS173: Discrete Math - University of California, Merced Thus, there are integers k and l with a-b=km and b-c=lm Put them together a-c=(a-b)+(b-c) = km+lm = (k+l)m Thus a≡c(mod m) So, congruence modulo m is transitive It follows that congruence modulo m is an equivalence relation * Example Let R be the relation on the set of real numbers s t xRy iff x and y are real numbers that differ by less
Introduction to Discrete Structures Introduction R1={ (a,b) | a b } R2={ (a,b) | a,b N, a b Z } R3={ (a,b) | a,b N, a-b=2 } R1 is reflexive since for every a N, a a R2 is reflexive since a a=1 is an integer R3 is not reflexive since a-a=0 for every a N Properties: Symmetry Definitions: A relation R on a set A is called symmetric if a,b A ( (b,a) R (a,b) R ) A relation R on a set A is called