COEN 231 Introduction to Discrete Mathematics Assignment 1 solution Concordia University

COEN 231 Introduction to Discrete Mathematics
Assignment 1 solution Concordia University

, 1) Suppose that p and q are statements so that p → q is false. Find the truth values of each of the
following:
p → q is false means p is true and q is false (refer to definition)

a. ⌐p → q p is True then ⌐p is False then ⌐p → q is true
b. p∨q p is true then p ∨ q is true
c. q →p q is False then q →p is True
2)
a. If the Truth value of (p→r)↔(q→r) is True and the truth value of p is True and r is
False, what is the Truth value of q.
the truth value of p is True and r is False (p→r) is False then (q→r) is false , r is False then q is
True
b. Draw the truth table and support your answer.
p q r p→r q→r (p→r)↔(q→r)
1 1 1 1 1 1
1 1 0 0 0 1
1 0 1 1 1 1
1 0 0 0 1 0
0 1 1 1 1 1
0 1 0 1 0 0
0 0 1 1 1 1
0 0 0 1 1 1
3) Determine whether the following statements are logically equivalent:
a. p→(q→r) and (p→q)→r
Assume
p is F q is T r is F
q is T then q→r is F and P is F p→(q→r) is T
p is F q is T (p→q) is T r is F then (p→q)→r is F
We find one example that two statements do not have the same truth value.

b. p → (q → r) and p → (q ∧ r)
Try these truth values, p is T q is F r is T show they are not equivalent.

c. (p→r)∨(q→r) and (p∧q)→r
(p → r) ∨ (q → r) ≡ (¬p ∨ r) ∨ (¬q ∨ r) definitionof implication
(¬p ∨ ¬q) ∨ r association and comulative law
≡ ¬(p ∧ q) ∨ r De Morgan
≡ (p ∧ q) → r definition of implication


4) Show that (p→q)→(r→s) and (p→r)→ (q → s) are not logically equivalent.
Consider
p q r s p→q r→s (p→q)→(r→s) p→r q → s (p→r)→ (q → s)
F T F F T T T T F F


5) Prove that the converse and inverse of a conditional statement are logically equivalent to each other.
a. Using Truth table
P Q ¬p ¬q ¬p → ¬q q→p

Concordia university, Department of ECE Manijeh Khataie