CMSC 250 Fall 2001 Homework 3 solutions

1. 1. Modus Tollens

   2. Disjunctive Syllogism

   3. Conjunctive Simplification

   4. Hypothetical Syllogism

2. The logical expression equivalent to the given circuit is: $(P \wedge Q) \vee \sim R$

3. 						1	2	3	4	5
   p	q	r	$\sim p$	$\sim q$	$\sim r$	$r \wedge q$	$p \wedge \sim q$	$(1) \rightarrow \sim p$	$p \rightarrow \sim r$	$p \vee r$
   T	T	T	F	F	F	T	F	F	F	T
   T	T	F	F	F	T	F	F	T	T	T
   T	F	T	F	T	F	F	T	T	F	T
   T	F	F	F	T	T	F	T	T	T	T
   F	T	T	T	F	F	T	F	T	T	T
   F	T	F	T	F	T	F	F	T	T	F
   F	F	T	T	T	F	F	F	T	T	T
   F	F	F	T	T	T	F	F	T	T	F

   The fourth row is the only critical row and since the conclusions in this row are true, we conclude that the argument is VALID.

4. 						1	2	3	4	5	6
   p	q	r	$\sim p$	$\sim q$	$\sim r$	$p \wedge \sim q$	$p \wedge r$	$(1) \vee (2)$	$\sim (3)$	$q \wedge \sim r$	$\sim p \vee (5)$
   T	T	T	F	F	F	F	T	T	F	F	F
   T	T	F	F	F	T	F	F	F	T	T	T
   T	F	T	F	T	F	T	T	T	F	F	F
   T	F	F	F	T	T	T	F	T	F	F	F
   F	T	T	T	F	F	F	F	F	T	F	T
   F	T	F	T	F	T	F	F	F	T	T	T
   F	F	T	T	T	F	F	F	F	T	F	T
   F	F	F	T	T	T	F	F	F	T	F	T

   Columns 4 and 6 are identical, Hence the two circuits are equivalent.

5. 1. $10011_2$

   2. $1010010_2$

   3. $87_{10}$

   4. $10000000000_2$

   5. $85_{10}$

   6. $5D_{16}$

   7. $1111101011011110_2$

6. Solutions to the binary addition are:

   1. 10001

   2. 10010000

   3. 1010000000100

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