1 Henry Selvaraj Henry Selvaraj Henry Selvaraj Henry

1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; truth Henrytable Selvaraj; (schematic capture, VHDL, and. Henry etc. ) Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; (traditionally into Boolean Henry Selvaraj; Henry Selvaraj; Henryexpressions) Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Traditional steps in a CAD system for implementing a digital circuit using PLD • design entry • translating into a standard form • logic minimization • technology mapping • placement and routing Henry Selvaraj

2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • decomposition chart technique (Ashenhurst 1959, Curtis 1962), Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • abstract decomposition (Roth - Karp 1962), Henry Selvaraj; Henry Selvaraj; • algebraic and. Selvaraj; Boolean division (Brayton Henry Selvaraj; methods Henry Selvaraj; Henry 1982, Selvaraj; Abouzeid Henry Selvaraj; Henry 1990, Selvaraj; Henry 1992), Selvaraj; Henry Selvaraj; Rajski Henry Selvaraj; Henry Selvaraj; • spectral techniques et al. 1985) Henry Selvaraj; (Hurst Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; (Brayton Henry Selvaraj; • multiple-level BDD Henry and other decision graph Henry approaches Henry Selvaraj; Henry Selvaraj; 1986) Henry Selvaraj; Henry Selvaraj; • residues based. Selvaraj; method (Mc. Cluskey 1986), Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • decomposition using groupability method (Bochmann et al. 1991), Henry Selvaraj; Henry Selvaraj; • functional decomposition Sasao 1993), Henry Selvaraj; Henry (Perkowski Selvaraj; Henry 1992, Selvaraj; Henry Selvaraj; Henry Selvaraj; • balanced multi-level functional decomposition using symbolic Henry Selvaraj; Henry Selvaraj; Some important logic minimization methods decomposition concept (Selvaraj, Luba 1994) Henry Selvaraj

3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Why decomposition? • Minimization techniques concentrate on minimizing the number of terms in an expression. • The only restriction imposed by FPGA is the number of inputs Henry Selvaraj

4 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; How. Henry to take it in? Decompose! Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; What is decomposition? Henry Selvaraj

5 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Henry Selvaraj; Henry k Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Ashenhurst, in his fundamental paper, stated the disjunctive decomposition theorem based on the notion of decomposition charts. Curtis extended the Ashenhurst's results to multiple decomposition when F is expressed as F = H(A, G (B), . . . , G (B)). They were the first to introduce the concept of functional decomposition. Henry Selvaraj

6 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; In the early 80's, functional decomposition methods lost their importance because of the rapid development of synthesis techniques referred to as division based methods. Algebraic division of sum-of-products expressions represented by the sets of cubes became a basic operation in the procedures of substitution and kernel extraction used for decomposition of Boolean functions Henry Selvaraj

7 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Since the late 80's logic decomposition has been again attracting some attention as a technique used for design of PLAs. Devadas et al. proposed Boolean decomposition of a PLA into two cascaded PLAs. Henry Selvaraj

8 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Functional decomposition can play an important role in the design of FPGA-based circuits because their structure imposes constraints on the number of inputs only and the two-level minimization is not needed. However, the division based synthesis became so deeply rooted that earlier synthesis methods for FPGAs were based on the multilevel minimization approach. Henry Selvaraj

9 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Recollection of useful definitions: A Boolean variable is a single co-ordinate in a Boolean space. A literal is a Boolean variable or its complement. A cube c is a set of literals such that x c x’ c. Two trivial cubes 0 and 1 exist. They are defined as the Boolean functions 0 and 1 respectively. A Boolean expression is a set of cubes. Henry Selvaraj

10 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Definitions contd. . A Boolean expression is called non redundant if no cube of the expression properly contains an another cube. For example, a + ab is redundant because {a} {a, b}. The expression a + b is not redundant. Henry Selvaraj

11 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Definitions contd. . The support is the set of Boolean variables which appear either complemented or uncomplemented in the expression f. Two functions f and g are said to have a disjoint support if and only if sup(f) sup(g) = . Henry Selvaraj

12 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Definitions contd. . An incompletely specified single-output function F is represented by a triplet of completely specified single-output functions: F(f, d, r), where f represents the ON-set, i. e. all input vectors for which F evaluates to 1. Similarly, d and r are representations of the don't care set and the OFF-set, respectively. Henry Selvaraj

13 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; i j. Henryi Selvaraj; Henry j Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Definitions contd. . The product of two expressions f and g (denoted f g) is defined as f g = {c d c f d g}. Henry Selvaraj

14 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; A Boolean function F of. Selvaraj; n binary variables x 1, . . xn. Henry can. Selvaraj; be Henry Selvaraj; Henry Selvaraj; expressed as: Selvaraj; F: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; care'. Henry Selvaraj; Henry Selvaraj; n {0, 1} Henry Selvaraj; Henryfunction Selvaraj; Henrym. Selvaraj; If the F is. Selvaraj; expressed as F: {0, 1} , Henry Selvaraj; Henry Selvaraj; then. Selvaraj; it is called a completely specified function. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; A function with a single output (i. e. m = 1), is called a unitary Henry Selvaraj; Henry Selvaraj; function is. Selvaraj; denoted by. Selvaraj; f. On. Henry the Selvaraj; other hand, if the. Henry number Henry Selvaraj; and Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; of outputs is more than one, the function is called a group of Henry Selvaraj; Henry Selvaraj; (unitary) and is Selvaraj; denoted by. Selvaraj; F. Henry Selvaraj; functions Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Basic Notions Henry Selvaraj

15 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; n is Henry An element of the set D called a minterm (sometimes Henry Selvaraj; Henry Selvaraj; Henry and Selvaraj; Henry Selvaraj; known as. Henry input vector) its. Henry respective output is Henry called an Henry Selvaraj; Henry Selvaraj; output vector. Henry Selvaraj; Henry Selvaraj; representation Henry Selvaraj; The tabular of. Selvaraj; minterms and their Henry Selvaraj; Henry Selvaraj; respective output vectors generated by the. Henry function is called Henry Selvaraj; Selvaraj; FHenry Selvaraj; Henry In Selvaraj; Henrytable Selvaraj; Henrynot Selvaraj; Henry Selvaraj; a truth table. general, the truth does contain input Henry Selvaraj; Henry Selvaraj; vectors v, Henry for which all the outputs 'do Selvaraj; not care' values. Henry Selvaraj; Henryassume Selvaraj; Henryauthors Selvaraj; Henry Selvaraj; Hence some define the Henry function F Henry simply as: Henry Selvaraj; Henry Selvaraj; m, where n. Henry Selvaraj; Henry Selvaraj; F: Henry Dn {0, 1} Dn Selvaraj; {0, 1}Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Basic notions contd. . . Henry Selvaraj

16 Basic notions contd. . . Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; The basic requirement of a multilevel logic synthesis is to Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry implement a function F Selvaraj; as a logic circuit, i. e. Selvaraj; into Henry a set. Selvaraj; of Henry Selvaraj; Henry n Selvaraj; m. Selvaraj; elements transforming the binary vectors D {0, 1, -, } Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; However, considering the wide spectrum of. Selvaraj; technological Henry Selvaraj; Henry Selvaraj; possibilities, it. Selvaraj; is not. Henry necessary to be more. Henry specific the Henry Selvaraj; about Henry Selvaraj; Henryof Selvaraj; Henry Selvaraj; expressions 'logic circuit' and 'set elements'. They. Henry can. Selvaraj; be Henry Selvaraj; Henry Selvaraj; understood according to the demand of the. Henry circumstances. But Henry Selvaraj; Selvaraj; Henry Selvaraj; Henry Selvaraj; can Henry be Selvaraj; it is. Selvaraj; important to realize that Henry an implementation a Henry Selvaraj; Henry Selvaraj; multilevel implementation also. Henry Hence concrete distinction of Henry Selvaraj; Henry Selvaraj; variables Henry Selvaraj; is Henry Selvaraj; only Henry Selvaraj; input and. Henry output valid at the initial Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; specification level. Henry Selvaraj

s 17 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Consider tuple T = Henry (M, A, X, Y), Henry Selvaraj; a. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Selvaraj; Henry Selvaraj; a: M VHenry A, VX = {0, 1}, VY = {0, 1, -}. Basic notions contd. . . where: M - non empty finite set of minterms. A - finite variables and Y is the set of output variables); A = X Y and X Y = , where for each a A, which a - function values for every minterm v M, i. e. : Henry Selvaraj

18 Basic notions contd. . . Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry As Selvaraj; every. Henry pair. Selvaraj; of minterms in Henry the Selvaraj; specification table TSelvaraj; may Henry Selvaraj; Henry Selvaraj; have certain inputs, convenient form of expressing Henry Selvaraj; Henryidentical Selvaraj; Henry Selvaraj; a. Henry Selvaraj; Henry Selvaraj; Selvaraj; such an 'identity' is Henry an indiscernibility relation. This. Henry relation, Henry Selvaraj; Henry Selvaraj; called IND be. Henry defined as follows for. Selvaraj; the Henry Selvaraj; Henry relation, Selvaraj; Henrycan Selvaraj; Henry of Selvaraj; Henry Selvaraj; requirements Boolean functions: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Definition: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Let B X and v , v 2 M. Minterms (v , v 2) belong to Henry Selvaraj; Henry 1 Selvaraj; Henry Selvaraj; if. Henry IND(B) and. Selvaraj; only if. Henry for. Selvaraj; every. Henry x Selvaraj; B, x(v. Henry x(v 2). Henry Selvaraj; 1) = Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

19 Basic notions contd. . . Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Therefore, the IND relation divides the set M into M/IND(B) Henry Selvaraj; Henry Selvaraj; abstract classes. In. Henry further analysis, for Henry simplicity Henry Selvaraj; equivalence Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; reasons, the abstract classes M/IND(B) are denoted as P(B) and Henry Selvaraj; Henry Selvaraj; called as Henry the input the set B. the Henry Selvaraj; partitions Henry Selvaraj; generated Henry Selvaraj; by Henry Selvaraj; Henry. So, Selvaraj; Henry Selvaraj; Henry Selvaraj; equation can be rewritten as: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; P(B) =Henry Selvaraj; P(x)Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; x B Henry Selvaraj; Henry Selvaraj; Selvaraj; Henry x Selvaraj; where denotes the. Henry product of. Henry input. Selvaraj; partitions for every B. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

20 Basic notions contd. . . Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Two output vectors p and q are said to be consistent if their Henry Selvaraj; Henry Selvaraj; respective components, defined, are equal: Henry Selvaraj; Henrywhich Selvaraj; are Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; i {n+1, . . . , n+m}, (p = q ) (p = -) (qi =Henry -) Selvaraj; Henry Selvaraj; i Henryi Selvaraj; i Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; is Henry Selvaraj; by Henry Consistency relation of the output vectors denoted p Selvaraj; ~ q. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; For. Selvaraj; example, p = (00 -), q = (-01) r =Henry (0 -0), it is. Henry said. Selvaraj; that p Henryif Selvaraj; Henry and Selvaraj; Henry Selvaraj; ~ q, Selvaraj; p ~ r. Henry and Selvaraj; q _ r (q is Selvaraj; inconsistent with Henry r). Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

21 Basic notions contd. . . Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry m Selvaraj; Every consistent pair of input vectors p, q (p, q {0, 1, -} ) has Henry Selvaraj; Henry Selvaraj; its product vector r Henry defined as. Henry r =Selvaraj; p q =Henry (r 1, . . . , r where the Henry Selvaraj; Selvaraj; n), Henry Selvaraj; Henry Selvaraj; component ri is: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 if. Henry pi, Selvaraj; qi are respectively 0, 0; Selvaraj; 0, -; -, 0; Henry Selvaraj; Henry Selvaraj; 1 if. Henry pi, Selvaraj; qi are respectively 1, 1; Selvaraj; 1, -; -, 1; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; - if. Henry pi, q. Selvaraj; -, -. Selvaraj; Henry Selvaraj; i are respectively Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; For. Henry example if p =Selvaraj; (0 -1 -)Henry and. Selvaraj; q = (--10), then Henry their. Selvaraj; Henry Selvaraj; Henry Selvaraj; product vector r = p q = Selvaraj; = (0 -10). Henry Selvaraj; Henry Selvaraj; Henry Selvaraj