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(xy) means “x and y are real numbers such that x 2y 5”.
Determine whether the statement is true.
6.
xyP(xy).
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.
xyP(xy).
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: pr qr
(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
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(xy) means “x and y are real numbers such that x 2y 5”.
Determine whether the statement is true.
6.
xyP(xy).
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.
xyP(xy).
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: pr qr
(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