Email us to get an instant 20% discount on highly effective K-12 Math & English kwizNET Programs!

#### Online Quiz (WorksheetABCD)

Questions Per Quiz = 2 4 6 8 10

### High School Mathematics - 214.2 Truth tables

A truth table is a convenient way of summarising the truth values of a statement. A truth table consists of rows and columns. The letter T is used to indicate the truth value of a true statement and the letter F is used to indicate the truth value of a false statement.

Truth nd Falsity of a conjunction
Example: Construct the truth table for p^q.
First, we have to list all the possible combinations of true and false for p and q.
There are 2 components p and q, there would be 22 = 4 rows.
 p q p ^ q T T -- T F -- F T -- F F --

Since we have agreed that p ^ q is true only when both p and q are true, the complete truth table is as shown below.
 p q p ^ q T T T T F F F T F F F F
a. Let us consider the conjunction 5+3 = 8 and 7-1= 4. Here the component 5+3 = 8 is true but the component 7-1 = 4 is false. Hence, the conjunction is false.
b. '3 is an odd number and 4 is an even number ' is a true statement, since both the components of this conjunction are true statements.
c. The statement 'A is hale and healthy' is true only if both the conditions are satisfied, i,e, A must be hale and A must be healthy. If either or both of these conditions is not satisfied, the conjunction is false statement.

Truth and Falsity of a disjunction
 p q p v q T T T T F T F T T F F F

If the letters p and q represent two simple statements, the the truth value is as shown above.

Truth table for a negation (~p)
Since ~p consists of only one simple statement p, so there are 21 = 2 rows in the truth table.
 p ~q T F F T
Example: Construct the truth table for the statement ~p v q.
1. Step 1: Write in the first two columns the four possible pairs of truth values for the statements p and q.
 p q ~p ~p v q T T -- -- T F -- -- F T -- -- F F -- --

2. Step 2: Using column 1 as reference, we negate the statement p.
 p q ~p ~p v q T T F -- T F F -- F T T -- F F T --

3. Step 3: Using columns 3 and 2 and recalling that a disjunction is false only when both components (3 and 2) are false, we enter the values for ~p v q in the column labelled 4. We note that the statement ~p v q is false only in row 2, because in this case both components (~p, q) are false. The completed truth table is as shown below.
 p q ~p ~p v q T T F T T F F F F T T T F F T T

Example: Construct the truth table for the statement ~(p^ -q)

1. Step 1: Write in the first two columns the four possible pairs of truth values for the statements p and q.
2. Step 2: Write the values of ~q in column 3.
3. Step 3: Recalling that a conjunction is true only when both its components are true, fill in the values or p ^ ~q in column 4. Now negate the values of p^~q and enter the values for ~(p^~q) in column 5.
 p q ~q (p^~q) ~(p^~q) T T F F T F F T T F F T F F T F F T F T

 Q 1: Construct truth table for (~p)^(~q)Answer: Q 2: Construct truth table for p v q.Answer: Q 3: p: It is cold, q: It is raining. Form the conjunction of p and q.(symbolically)p > qp^qp < q Q 4: p: It is cold, q: It is raining. Which of these is the disjunction of p and q.(symbolically)It is cold or it is raining.It is cold and it is raining. Q 5: Construct truth able for p^(~p). Answer: Q 6: Construct truth table for (p^q)^(~p).Answer: Q 7: Let p: It is raining today, q: There are 20 chairs in this room, give a verbal sentence which describes ~p.It is not raining today.It is raining and there are 20 chairs in this room. Q 8: Construct truth table for ~[p v (~q)]Answer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

#### Subscription to kwizNET Learning System offers the following benefits:

• Unrestricted access to grade appropriate lessons, quizzes, & printable worksheets
• Instant scoring of online quizzes
• Progress tracking and award certificates to keep your student motivated
• Unlimited practice with auto-generated 'WIZ MATH' quizzes