truth table for the compound statement q Ar → [(r Vp) ^ (q V p)].

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Truth table: Please make sure the truth table is correct and it has no mistakes. Correct the truth table if there are mistakes.
truth table for the compound statement q ^ r → [(r V p) ^ (q V p)].
Transcribed Image Text:truth table for the compound statement q ^ r → [(r V p) ^ (q V p)].
Step 1
To draw a truth table for the given compound statement, we need to
consider all possible combinations of truth values for the propositional
variables q, r, and p. Since there are three variables, there will be 2^3 = 8
possible combinations of truth values.
Step 2
To create the truth table, we can start by listing all possible
combinations of truth values for q, r, and p, and then evaluate the
compound statement for each combination. The table will have
columns for each variable and the compound statement, and rows for
each possible combination of truth values.
The truth table for the given compound statement is:
q|r|p|rvp|qvp | (rvp)^(qvp) | q^r|q^r-> [(rv p)^(q vp)]
E
TTF
FFFL
LLL
FTT
F
F|F|T|
F|F|
E
TFTF
TFFTTF
T|
T
I T
T
| T | F
T|
|
→
Т T
F
FELE
| T | T
F
|
T
FTFE
F
I
1
|
|
| F |
FFFFFFFF
T
T
T
T
T
T
T
T
Step 3
To evaluate the compound statement for each row,
We first evaluate the truth values of rvp and qvp, and,
Then evaluate (r v p)^(q v p) using the truth values we just obtained.
Finally, we evaluate q^r and q^r-> [(r v p)^(q v p)] using the truth values
we obtained for q^r and (r v p)^(q v p), respectively.
Overall, the final column shows the truth value of the entire compound
statement for each combination of truth values. Here, the compound
statement is true for all possible combinations of truth values, which
means it is a tautology.
Transcribed Image Text:Step 1 To draw a truth table for the given compound statement, we need to consider all possible combinations of truth values for the propositional variables q, r, and p. Since there are three variables, there will be 2^3 = 8 possible combinations of truth values. Step 2 To create the truth table, we can start by listing all possible combinations of truth values for q, r, and p, and then evaluate the compound statement for each combination. The table will have columns for each variable and the compound statement, and rows for each possible combination of truth values. The truth table for the given compound statement is: q|r|p|rvp|qvp | (rvp)^(qvp) | q^r|q^r-> [(rv p)^(q vp)] E TTF FFFL LLL FTT F F|F|T| F|F| E TFTF TFFTTF T| T I T T | T | F T| | → Т T F FELE | T | T F | T FTFE F I 1 | | | F | FFFFFFFF T T T T T T T T Step 3 To evaluate the compound statement for each row, We first evaluate the truth values of rvp and qvp, and, Then evaluate (r v p)^(q v p) using the truth values we just obtained. Finally, we evaluate q^r and q^r-> [(r v p)^(q v p)] using the truth values we obtained for q^r and (r v p)^(q v p), respectively. Overall, the final column shows the truth value of the entire compound statement for each combination of truth values. Here, the compound statement is true for all possible combinations of truth values, which means it is a tautology.
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