This was adjacency matrix after k=x and k=1 steps. Notice the shorter (green) path from x to 4.

This was adjacency matrix after k=x and k=1 steps. Notice the shorter (green) path from x to 4.

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## This was adjacency matrix after k=x and k=1 steps. Notice the shorter (green) path from x to 4.

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**1**3 2 5 1 2 1 x 2 This was adjacency matrix after k=x and k=1 steps. Notice the shorter (green) path from x to 4. Now use k=3 and see how we pick it up. We do get to look at paths of length > 2! 4 4 3 3 X 1 2 3 4 X 1 2 3 4 A[i,3]+A[3,j] < A[i,j] ?**7.2 Logic circuits**• Boolean Expressions • Logic Networks • Truth Functions • Can every truth table be written as a Boolean expression? As a computer circuit?**Switches**Control (off=false) + - Control (on=true) + -**A Or B**A + - B**A and B**A B + -**Not A**A - +**And, Or and Not**And ( . ) Or ( + ) Not (‘)**Boolean Expressions**• A Boolean Expression in n variables x1,x2, .. Xn, is any string of finite symbols formed by the following rules • 1. x1,x2, .. Xn are Boolean expressions • 2. If P and Q are Boolean Expressions, then so are (P+Q), PQ , and P’.**Boolean Expressions**(x + y) x + y’(xy+z) is a Boolean expression.**Truth Functions**• A truth function is a truth table for Boolean expressions**Find a network for xy’+z**X Y Z Exercise: Find a network for (xy’+z)’**Find a Boolean expression**x y ? z**We can do the following**Truth function Boolean expression Boolean expression logic network Can we find a Boolean expression for any truth function? Yes!**Canonical Form**• For the truth function shown, find a Boolean expression: True for: xyz’ xy’z x’yz x’yz’ x’y’z**So the truth function is given by:**xyz’+xy’z+x’yz+x’yz’+x’y’z This is the Canonical form for the truth function (a standard form into which we can place any two Boolean expressions for comparison.)**Simplify, using Boolean algebra:**xyz’+xy’z+x’yz+x’yz’+x’y’z = xyz’+xy’z+x’yz+x’yz’+x’y’z+x’yz’ =y’z(x+x’)+x’y(z+z’)+yz’(x+x’) =y’z+x’y+yz’ Notice the groupings of two the same/ one different.**Half Adder**• Design a circuit to add binary numbers**Half Adder-as two functions**xy’+x’y=(x+y)(xy)’ xy**The half-adder**(x+y)(xy)’ x sum y x y carry xy**To add two binary numbers**Full adder Cprevious s Half adder x c s cnext Half Adder y c**General add: xyz+abc**x Digit 1 Half adder a Digit 2 Full adder y b Digit 3 Full adder z c Digit 4