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High School Mathematics - 2
14.2 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 qp ^ q
TT--
TF--
FT--
FF--

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 qp ^ q
TTT
TFF
FTF
FFF
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 qp v q
TTT
TFT
FTT
FFF

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
TF
FT
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
    TT----
    TF----
    FT----
    FF----

  2. Step 2: Using column 1 as reference, we negate the statement p.
    p q~p ~p v q
    TTF--
    TFF--
    FTT--
    FFT--

  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
    TTFT
    TFFF
    FTTT
    FFTT

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)
    TTFFT
    FFTTF
    FTFFT
    FFTFT


Q 1: Construct truth table for p v q.
Answer:

Q 2: p: It is cold, q: It is raining. Form the conjunction of p and q.(symbolically)
p < q
p > q
p^q

Q 3:
Answer:

Q 4: Construct truth table for (p^q)^(~p).
Answer:

Q 5: Construct truth table for (~p)^(~q)
Answer:

Q 6: 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 7: Construct truth table for (pvq)v(~p)
Answer:

Q 8: Construct truth able for p^(~p).

Answer:

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