Consider the following propositions: • D: The door is locked. • L: The light is
ID: 3282779 • Letter: C
Question
Consider the following propositions:
• D: The door is locked.
• L: The light is on.
• S: The store is open.
Requirements:
Your submission must be your original work. No more than a combined total of 30% of the submission and no more than a 10% match to any one individual source can be directly quoted or closely paraphrased from sources, even if cited correctly. Use the Turnitin Originality Report available in Taskstream as a guide for this measure of originality.
You must use the rubric to direct the creation of your submission because it provides detailed criteria that will be used to evaluate your work. Each requirement below may be evaluated by more than one rubric aspect. The rubric aspect titles may contain hyperlinks to relevant portions of the course.
A. Express each of the following statements using only the logical operators ?, ?, ¬ , and ?; the propositional variables D, L, and S; and parentheses if needed:
1. If the store is open, then the light is on.
2. The store is open if the light is on and the door is not locked.
3. If the store is open, then the door being unlocked implies that the store is open.
4. If the store is closed, then the door is locked or the light is off.
B. Create a truth table for each of the four compound propositions from part A and fill in all truth values in each row of each table.
Note: In the truth table, include intermediate columns for all smaller parts of the proposition.
C. Identify which two of the four compound propositions in part A are logically equivalent.
1. Explain (suggested length of 1 sentence) how you know the two propositions are logically equivalent.
D. Explain what a tautology is, using the properties of logic.
1. Identify which, if any, of the four compound propositions in part A is a tautology.
E. Explain what a contradiction is, using the properties of logic.
1. Identify which, if any, of the four compound propositions in part A is a contradiction.
Explanation / Answer
A)
B)
i)
S
L
S ? L
T
T
T
T
F
F
F
T
T
F
F
T
ii)
L
D
S
¬D
L ? ¬D
(L ? ¬D) ? S
T
T
T
F
F
T
T
T
F
F
F
T
T
F
T
T
T
T
T
F
F
T
T
F
F
T
T
F
F
T
F
T
F
F
F
T
F
F
T
T
F
T
F
F
F
T
F
T
iii)
S
D
¬D
S ? ¬D
(S ? ¬D) ? S
T
T
F
F
T
T
F
T
T
T
F
T
F
T
F
F
F
T
T
F
iv)
L
D
S
¬S
¬L
D ? ¬L
¬S ? (D ? ¬L)
T
T
T
F
F
T
T
T
T
F
T
F
T
T
T
F
T
F
F
F
T
T
F
F
T
F
F
F
F
T
T
F
T
T
T
F
T
F
T
T
T
T
F
F
T
F
T
T
T
F
F
F
T
T
T
T
C) A tautology is a statement that is always true, no matter what
From above truth tables (ii) and (iv) are logically equivalent because their truth tables are same
D) The opposite of a tautology is a contradiction, a formula which is "always false". In other words, a contradiction is false for every assignment of truth values to its simple components.
From above truth tables (i) and (iii) are logically not equivalent because their truth tables are not same
S
L
S ? L
T
T
T
T
F
F
F
T
T
F
F
T