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Conjunctions and Disjunctions

Group #2III-Boyle

Conjunctions

Connectives are often used to join two simple statements are

and, or, if … .then, if and only if.

Two simple statements joined by the word “and is called

a conjunction. The symbol “^” is used to represent the

word “and”.

Conjunctions

Let p and q be simple statements.

p: The number 3 is an even integer

q: The number 7 is prime.

Then the following in conjunction symbols:

The number 3 is an even integer and the number 7 is prime.

P ^ Q

The conjunction is a true statement if only

both parts are true statements. Since p is

false statement, the conjunction is false. The

conjunction is usually expressed as and,

however, it can also be expressed as but,

however or nevertheless.

P Q P ^ Q

T T T

T F F

F T F

F F F

Conjunctions

Examples:

Consider the following statements.

a. Albay is in Bicol and 5+5=10

b. Albay is in Bicol and 5+5=11

c. Albay is in Laguna and 5+5=10

d. Albay is in Laguna and 5+5=12

Solution:

Only statement a is true. Each of the other statement is false

since at least one of its simple statements is false.

Conjunctions

Another Example

Given the truth values of p and q, state the truth value of the

indicated statement.

a. p true, q false; ~p ^ q b. p false, q false ~(~p ^

~q)Solution:

A.

B.

P Q ~P ~P ^ Q

T F F F

P Q ~P ~Q ~P ^ Q ~(~P ^ Q~)

F F T T T F

Conjunctions

Check Your Understanding

Which statement is true?

1. Chocolate Hills is in Bohol and 2 – 4 = 0

2. Chocolate Hills is in Iloilo and 2 – 4 = -2

3. Chocolate Hills is in Bohol and 2 – 4 = -2

Given the truth values of p and q, state the truth values of the

indicated statements

4. p true, q true; p ^ ~q 6. p false, q false; ~(p ^ q)

5. p true, q false; ~p ^ q 7. p false; q true; ~(~p ^ q)

Two simple statements joined by the word “or” is called a

disjunction. The symbol “v” is used to represent the word

“or”.

Disjunctions

Disjunctions

The disjunction of statements p and q giben in the previous page is

The number 3 is an even integer and the number 7 is prime.

P v Q

The above statement is true when at least

one of the parts is true. Since q is true, even

though p is false, p v q is true.

The truth table for disjunction is given at the

right.

P Q P ^ Q

T T T

T F T

F T T

F F F

Disjunctions

Examples:

Consider the following statements.

a. Albay is in Bicol or 5+5=10

b. Albay is in Bicol or 5+5=11

c. Albay is in Laguna or 5+5=10

d. Albay is in Laguna or 5+5=12

Solution:

Only statement d is false. Each of the other statement is true

since at least one of its simple statements is true.

Disjunctions

Another example

Let p and q be statements given by:

P: Triangle ABC has three sides.

Q: Triangle ABC has sides of the same length.

Triangle ABC has three sides Triangle ABC has sides of the same length

Write the following in symbolic form.

a. Triangle ABC has three sides and its sides are of the same length.

b. Triangle ABC has three sides or its sides are of the same length.

c. Triangle ABC has three sides and its sides are not of the same

length.

d. It is no true that triangle ABC has three sides and its sides are of the

same length.

Disjunctions

Another example:

Write the following compound statements in terms of the given

statements and give the truth value of the statement

P: 6 is even Q: 9 < 5

a. p ^ q

b. p v q

c. p ^ ~q

d. ~p v q

Disjunctions

Check Your Understanding

Write the following compound statements in terms of the given simple

statements and give the truth value of the statement.

M: 4 + 2 = 42 N: 4 x 2 > 5 O: 7 – 2 <0

1. M ^ N

2. ~N V O

3. M V O

4. ~M ^ O

ANY QUESTIONS???

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