Conditional Propositions and Logical Equivalence

Prepared by: Engr. Ma. Kristina P. BorbonPrepared by: Engr. Ma. Kristina P. Borbon

June 21, 2010

� The dean has announced that

If the Mathematics Department gets an additional Php 1M,

then it will hire one new faculty member.

� This proposition is called a conditional proposition.

Definition 1� If p and q are propositions, the compound proposition

If p and q

is called a conditional proposition and is denoted

p � qp � q

� The proposition p is called hypothesis (or antecedent) and the proposition q is called the conclusion (or consequent)

Example 1� If we define

p: If the Mathematics Department gets an additional Php 1M.

q: The Mathematics Department hires one new faculty q: The Mathematics Department hires one new faculty member.

� p is the hypothesis statement .

� q is the conclusion statement.

Example 2� Restate each proposition in the form of a conditional

proposition.

a) Mary will be a good student if she studies hard.

Ans: If Mary studies hard, then she will be a good student.Ans: If Mary studies hard, then she will be a good student.

b) John may take calculus only if he has sophomore, junior, or senior standing.

Ans: If John takes calculus, then he has sophomore, junior, or senior standing.

c) When you sing, my eyes hurt.

Ans: If you sing, then my eyes hurt.

d) A necessary condition for the Cubs to win the World Series is that they sign a right-handed relief pitcher.

Ans: If the Cubs win the World Series, then they sign a right-handed relief pitcher.a right-handed relief pitcher.

e) A sufficient condition for Ralph to visit California is that he goes to Disneyland.

Ans: If Ralph goes to Disneyland, then he visits California.

Definition 2� The truth table for the conditional proposition p � q:

If the Mathematics Department gets an additional Php 1M,

then it will hire one new faculty member.

Example 2� Let

p: 1 > 2, q: 4 < 8

then p is false and q is true. Therefore,

p � q is true, q � p is falsep � q is true, q � p is false

Example 3� Assuming that p is true, q is false, and r is true, find

the truth value of each proposition.

a) (p Λ q) � r

b) (p Ѵ q) � r’ b) (p Ѵ q) � r’

c) p Λ (q � r)

d) p � (q � r)

Example 3 - solution� We replace each symbol p, q, and r by its truth value to

obtain the truth value of the proposition:

a) (T Λ F) � T = F � T = true

b) (T Ѵ F) � T’ = T � F = falseb) (T Ѵ F) � T’ = T � F = false

c) T Λ (F � T) = T Λ T = true

d) T � (F� T) = T � T = true

� Logic is concerned with the form of propositions and the relation of propositions to each other and not with the subject matter itself.

� The proposition p � q can be true while the � The proposition p � q can be true while the proposition q � p is false.

� q � p is the converse of p � q.

� Thus, a conditional proposition can be true while its converse is false.

Example 4� Write each conditional proposition symbolically. Write

the converse of each statement symbolically and in words. Also, find the truth value of each conditional proposition and its converse.proposition and its converse.

a) If 1 < 2, then 3 < 6.

b) If 1 > 2, then 3 < 6.

Example 4 - solutiona) Let

p: 1 < 2, q: 3 < 6.

- Symbolically, p � q

- Since p and q are both true, this statement is true.- Since p and q are both true, this statement is true.

- Converse: symbolically, q � p

- In words: if 3 < 6, then 1 < 2.

- Since p and q are both true, the converse q � p is true.

Example 4 - solutionb) Let

p: 1 > 2, q: 3 < 6.

- Symbolically, p � q

- Since p is false and q is true, this statement is true.- Since p is false and q is true, this statement is true.

- Converse: symbolically, q � p

- In words: if 3 < 6, then 1 > 2.

- Since q is true and p is false, the converse q � p is false.

Definition 3� If p and q are propositions, the compound

proposition

p if and only if qp if and only if q

is called a biconditional proposition and is denoted

p �� q

� The truth table of the proposition p �� q:

� “ p if and only if q” � “p is a necessary and sufficient condition for q.”

� “ p if and only if q” � “p iff q”

Example 5� The statement

1 < 5 if and only if 2 < 8

� Symbolically: p �� q

� If we define

p: 1 < 5, q: 2 < 8

� Since both p and q are true, p �� q is true.

� Alternative statement: A necessary and sufficient condition

for 1 < 5 is that 2 < 8.

� In some cases, two different compoundpropositions have the same truth values nomatter what the truth values theirmatter what the truth values theirconstituent propositions have.

�This propositions are said to be

logically equivalent.

Definition 4� Suppose that the compound propositions P and Q are

made up of the propositions p1,. . ., pn.

� We say that P and Q are logically equivalent and write

P ≡ Q,P ≡ Q,

provided that given any truth values of p1,. . ., pn, either Pand Q are both true or P and Q are both false.

De Morgan’s Laws for Logic(p Ѵ q)’ ≡ p’ Λ q’

(p Λ q)’ ≡ p’ Ѵ q’

� Truth table:

Thus P and Q are logically equivalent.

Example 6� Show that the negation of p � q is logically equivalent

p Λ q’.

(p � q)’ ≡ p Λ q’

Truth table:Truth table:

Thru this Truth Table, we can verify that given any truth values of p and q, either P and Q are both true or P and Qare both false.

Example 7� p �� q ≡ (p � q) Λ (q � p)

� Truth table:

Definition 5� Contrapositive – is an alternative, logically equivalent

form of the conditional proposition.

� The contrapositive (or transposition) of the � The contrapositive (or transposition) of the conditional proposition p � q is the proposition

q’ � p’ .

Example 8�Write the proposition

If 1 < 4, then 5 > 8

symbolically. Write the converse and symbolically. Write the converse and contrapositive both symbolically and in words. Find the truth table of each proposition.

Example 8 - solution� If we define

p: 1 < 4, q: 5 > 8

�Proposition: p � q �Proposition: p � q

�Converse: q � p

� In words: If 5 > 8, then 1 < 4.

�Contrapositive: q’ � p’

� In words: if 5 is not greater than 8, then 1 is not less than 4.

Example 8 - solution� p � q is false, q � p is true

� q � p is false.

� An important fact is that a conditional proposition and its contrapositive are logically equivalent.contrapositive are logically equivalent.

� Truth Table:

Assignment (1 whole)In exercises 1-4, restate each proposition in the form of a

conditional proposition:

1. Joey will pass the discrete mathematics exam if he studies hard.hard.

2. A sufficient condition for Katrina to take the algorithms course is that she pass discrete mathematics.

3. The program is readable only if it is well structured.

4. Rosa may graduate is she has 160 quarter-hours of credits.

AssignmentIn exercises 5-8, assuming that p and r are false and that

q and s are true, find the truth value of each proposition:

5. p � q 5. p � q

6. (p � q)’

7. (p � q ) Λ (q � r)

8. (s �(p Λ r’)) Λ((p � (r Ѵ q)) Λ s)

AssignmentIn exercises 9 -11, write each conditional proposition

symbolically. Write the converse and contrapositive of each statement symbolically and in words. Also, find the truth value of each conditional proposition, its the truth value of each conditional proposition, its converse, and its contrapositive

9. If 4 < 6, then 9 > 12.

10. If 4 > 6, then 9 > 12.

11. |4| < 3 if -3 < 4 < 3.

AssignmentIn exercises 12-15, for each pair of propositions P and Q,

state whether or not P ≡Q. 12. P = p, Q = p Ѵ q

13. P = p Λ q, Q = p’ Ѵ q’13. P = p Λ q, Q = p’ Ѵ q’

14. P = p Λ (q’ Ѵ r), Q = p Ѵ (q Λ r’)

15. P = p � q, Q = p�� q