eci 114

UC Davis, ECS 20, Winter 2014
Instructor: Vladimir Filkov

Midterm Review Exercises

1.

Write the truth table for the proposition (r  q)  (p  r). q Ans: p  q)  (p  r) r (r

p
T
T
T
T
F
F
F
F
2.

T
T
F
F
T
T
F
F

T
F
T
F
T
F
T
F

T
T
F
T
T
F
F
F

Find a proposition with three variables p, q, and r that is true when exactly one of the three variables is true, and false otherwise
Ans: (p  q  r)  (p  q  r)  (p  q  r).

3.

Determine whether p  (q  r) and p  (q  r) are equivalent.
Ans: Not equivalent. Let q be false and p and r be true.

4.

Write the contrapositive, converse, and inverse of the following: You sleep late if it is Saturday.
Ans: Contrapositive: If you do not sleep late, then it is not Saturday. Converse: If you sleep late, then it is Saturday. Inverse: If it is not Saturday, then you do not sleep late.

5.

On the island of knights and knaves you encounter two people. A and B. Person A says, "B is a knave." Person B says, "At least one of us is a knight." Determine whether each person is a knight or a knave.
Ans: A is a knave, B is a knight.

In the two questions below P(xy) means “x and y are real numbers such that x  2y  5”.
Determine whether the statement is true.
6.

xyP(xy).
Ans: True, since for every real number x we can find a real number y such that x
 2y  5, namely y  (5  x)2.

7.

xyP(xy).
Ans: False, if it were true for some number x0, then x0 = 5 -2y for every y, which is not possible.

8.

Determine whether the following argument is valid: pr qr
(p  q)
________
 r
Ans: Not valid: p false, q false, r true

9.

Prove that the following is true for all positive integers n: n is even if and only if
3n2  8 is even.
Ans: If n is even, then n  2k. Therefore 3n2  8  3(2k)2  8  12k2  8  2(6k2 
4), which is even. If n is odd, then n  2k  1. Therefore 3n2  8  3(2k 
1)2  8  12k2  12k