discrete mathematics for a computer programming course Show

(discrete mathematics for a computer programming course)

Show that: (a) (p rightarrow q) (q rightarrow r) rightarrow (p rightarrow r) is a tautology. (b) (p rightarrow q) rightarrow r and p rightarrow (q rightarrow r) are not logically equivalent. (c) p doubleheadarrow q and not p doubleheadarrow not q are logically equivalent.

Solution

And the last column i.e., (p->q) ^ (q->r) -> (p->r) shows it is a tautology.

And the last two column i.e., (p->q)->r, and p->(q->r) are not the same in all the rows. And therefore, they are not logically equivalent.

And the columns (3), and (6) i.e., (p<->q), and (~p<->~q) are the same in all the rows. And therefore, they are logically equivalent

P(1)	Q(2)	R(3)	P->Q(4)	Q->R(5)	P->R(6)	4^5 (7)	7->6
0	0	0	1	1	1	1	1
0	0	1	1	1	1	1	1
0	1	0	1	0	1	0	1
0	1	1	1	1	1	1	1
1	0	0	0	1	0	0	1
1	0	1	0	1	1	0	1
1	1	0	1	0	0	0	1
1	1	1	1	1	1	1	1