1. Using the following declarations, determine
if the following statements are true or false.
int j = 4; int k = 8;
int y = 9;
(a) j <= 9
(b) y <= k
(c) (y+1 > j)
(d) (y < k) && (y < j)
(e) (k > j) || ((j < 3) && (j > 1))
(f) !(k < j)
(g) ! ((k > 9) || (k > 1))
(h) ! (!(y>2) && !(k<2))
2. Using the following declarations, determine
if the following statements are true or false.
int w = 0; int h = 0;
int r = 1;
(a) w < 0
(b) h < w
(c) h <= w
(d) !(h == r)
(e) !(h < r)
(f) !((r > h) && (w <= h))
(g) ((r > w) && (w < 4)) || (w > -1)
(h) (r < 4) || ((w > h) || !(r < 3))
3. Prove the following Identities by filling in the truth tables.
(a) X || X = X
| X | X||X | X |
| False | ||
| True |
(b) X && X = X
| X | X&&X | X |
| False | ||
| True |
(c) !(X||Y) = !X && !Y
| X | Y | !(X||Y) | !X && !Y |
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |
(d) !(X&&Y) = !X || !Y
| X | Y | !(X&&Y) | !X || !Y |
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |
(e) X && (X||Y) = X
| X | Y | X && (X||Y) |
| T | T | |
| T | F | |
| F | T | |
| F | F |
(f) X && (X||Y) = X || (X&&Y)
| X | Y | X && (X||Y) | X || (X&&Y) |
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |
(g) X && (!X || Y) = (!X && X) || (X && Y)
| X | Y | X && (!X || Y) | (!X && X) || (X && Y) |
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |
(h) (X || Y) && (X || !Y) = X || (Y && !Y)
| X | Y | (X || Y) && (X || !Y) | X || (Y && !Y) |
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |