Exercise 2: POS Form for XOR — Possible Solution ==================================================================== GIVEN ------------------------------ Same XOR function: f(x,y)=1 for (0,1) and (1,0); f(x,y)=0 for (0,0) and (1,1) STEP 1: IDENTIFY THE 0-ROWS AND THEIR MAXTERMS ------------------------------ Row x=0,y=0: maxterm is x+y (only 0 when x=0 AND y=0) Row x=1,y=1: maxterm is x'+y' (only 0 when x=1 AND y=1) STEP 2: AND THE MAXTERMS TOGETHER ------------------------------ POS form: (x+y)(x'+y') STEP 3: VERIFY AGAINST THE ACTUAL XOR TABLE ------------------------------ x=0,y=0: (x+y)=0+0=0, (x'+y')=1+1=1. POS = 0.1 = 0. XOR(0,0)=0. Match. x=0,y=1: (x+y)=0+1=1, (x'+y')=1+0=1. POS = 1.1 = 1. XOR(0,1)=1. Match. x=1,y=0: (x+y)=1+0=1, (x'+y')=0+1=1. POS = 1.1 = 1. XOR(1,0)=1. Match. x=1,y=1: (x+y)=1+1=1, (x'+y')=0+0=0. POS = 1.0 = 0. XOR(1,1)=0. Match. RESULT ------------------------------ POS form of XOR: (x+y)(x'+y') - verified to match the real XOR truth table for all 4 input combinations, exactly matching Exercise 1's own SOP form (x'y + xy') on every input, just built via the opposite (maxterm/AND) construction. WHY THIS WORKS AS AN ANSWER ------------------------------ The maxterm for each 0-row is derived individually (using plain, not complemented, variables for the ones that were 0 in that row - the maxterm rule is the mirror image of the minterm rule), AND-ed together per this chapter's own POS procedure, and every one of the 4 possible inputs is checked against the actual XOR definition.