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# Solve 8 Propositional Logic Problems

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Logic

1. Let p, q, and r be the following statements:
p: Roses are red
q: The sky is blue
r: The grass is green
Translate the following statements into English
(a) p &#61657; q (b) p &#61657; (q &#61658; r) (c) q &#61614; (p &#61657; r) (d) ( &#61566; r &#61657; &#61566; q) &#61614; &#61566; p

2. Write in symbolic form using p, q, r, &#61566;, &#61614;, &#61658;, &#61657;, where p, q, r represent the following statements:
p: A puppy is green-eyed
q: A puppy can be taught
r: A puppy loves toys

(a) If a puppy is green-eyed, then it cannot be taught
(b) If a puppy cannot be taught, then it does not love toys
(c) If a puppy loves toys, then either the puppy can be taught or the puppy is green-eyed.
(d) If the puppy is not green-eyed, then the puppy loves toys and the puppy can be taught.

3. Fill the headings of the following truth table using p, q, &#61566;, &#61614;, &#61658;, and &#61657;.

p q (a) (b)
T T T F
T F T F
F T T F
F F F T

4. For each of the following conditionals, identify the antecedent and the consequent. Form the converse, inverse, and contrapositive.

(a) If I don't go to the movie, I'll study my math.
(b) Your car won't start if you don't have gasoline in the tank.

5. Use De Morgan's laws to write an equivalent statement for the following sentence:
If we go to San Antonio, then we will go to Sea World or we will not go to Busch Gardens.

6. (a) Translate the argument into symbolic form and (b) determine if the argument is valid or invalid. You may compare the argument to a standard form or use a truth table.

If Lillian passes the bar exam, then she will practice law.
Lillian will not practice law
----------------------------------------------------------------------
&#61532; Lillian did not pass the bar exam

7. Use an Euler diagram to determine whether the syllogism is valid or invalid
All actresses are beautiful
Some actresses are tall
--------------------------------
&#61532; Some beautiful people are tall.
8. Construct a truth table for (&#61566;q&#61657;p) &#61614; &#61566;q

https://brainmass.com/math/discrete-math/solve-8-propositional-logic-problems-113011

#### Solution Preview

Logic

1. Let p, q, and r be the following statements:
p: Roses are red
q: The sky is blue
r: The grass is green
Translate the following statements into English
(a) p &#61657; q (b) p &#61657; (q &#61658; r) (c) q &#61614; (p &#61657; r) (d) ( &#61566; r &#61657; &#61566; q) &#61614; &#61566; p

a. Roses are red and the sky is blue.
b. Roses are red, and either the sky is blue or the grass is green.
c. If the sky is blue, then roses are red and the grass is green.
d. If the grass isn't green, and the sky isn't blue, then roses aren't red.

Explanation: replace each letter with the appropriate statement. If there is a ~ in front, change the statement to its opposite. &#61657; means "and", and &#61658; means "or" - the way to remember which is which is that &#61657; sort of looks like a capital A, which stands for "and". &#61614; means if <thing on the left>, then <thing on the right>. In resolving things into understandable sentences, work your way from inside parentheses to outside. Resolve ~ first, then &#61657; and &#61658;, then &#61614;. Explanation of (b), step by step:

Roses are red &#61657; (The sky is blue &#61658; The grass is green)
Roses are red &#61657; (The sky is blue or The grass is green)
Roses are red and (The sky is blue or The grass is green)
Roses are red, and either the sky is blue or the grass is green.

2. Write in symbolic form using p, q, r, &#61566;, &#61614;, &#61658;, &#61657;, where p, q, r represent the following statements:
p: A puppy is green-eyed
q: A puppy can be taught
r: A puppy loves toys

(a) If a puppy is green-eyed, then it cannot be taught
(b) If a puppy cannot be taught, then it does not love toys
(c) If a puppy loves toys, then either the puppy can be taught or the puppy is green-eyed.
(d) If the puppy is not green-eyed, then the puppy loves toys and the puppy can be taught.