LESSON 2.2 Compound Propositions Flashcards
What is a simple proposition?
A simple proposition is a proposition with only one subject and only one predicate.
For example, “Every cat that barks has a PhD” is a simple proposition where “every cat that barks” is the subject, and “has a PhD” is the predicate.
How are compound propositions formed in logic?
Compound propositions are formed by combining simple propositions using logical connectives such as “or,” “and,” “but,” and “unless.”
What is the conjunction of two propositions p and q?
The conjunction of two propositions p and q is the proposition “p and q,” denoted by p ∧ q. It is true only when both p and q are true. If either one is false, the conjunction is false.
When is the conjunction of two propositions p ∧ q true or false?
The conjunction p ∧ q (“p and q”) is true only when both p and q are true. If either p or q (or both) are false, then the conjunction p ∧ q is false.
What is the disjunction of two propositions p and q?
The disjunction of propositions p and q is the proposition “p or q,” denoted by p ∨ q. It is false only when both p and q are false.
When is the disjunction p ∨ q true?
The disjunction p ∨ q is true if at least one of p or q is true (or if both are true).
What is a conditional statement p → q?
A conditional statement p → q is the proposition “If p, then q.” It is false only when p is true and q is false.
What are the converse, inverse, and contrapositive of p → q?
The converse is q → p, the inverse is (¬ p) → (¬ q), and the contrapositive is (¬ q) → (¬ p).
What is the truth table for p → q?
|p |1 1 0 0
| q |1 0 1 0
| p → q |1 0 1 1
In a conditional statement p → q, what are p and q called?
In the proposition p → q, p is called the premise and q is called the conclusion.
When is a conditional statement trivially true?
A conditional statement is trivially true when the premise p is false.
Give an example of a conditional statement with its premise and conclusion.
Example: “If you don’t wash the dishes, then you don’t get money for a buffet.” The premise is “You don’t wash the dishes,” and the conclusion is “You don’t get money for a buffet.”
Under what condition is the statement “If you don’t wash the dishes, then you don’t get money for a buffet” false?
The statement is false only when you don’t wash the dishes, but you still get money for the buffet.
What are other ways to express the conditional statement p → q?
Other ways to express p → q include “q if p,” “p implies q,” “p is sufficient for q,” or “q is necessary for p.”
Give an example of p, q, and p → q with their converse, inverse, and contrapositive.
Let p be “π is irrational” and q be “3 is less than 2.”
- p → q: If π is irrational, then 3 is less than 2.
- Converse (q → p): If 3 is less than 2, then π is irrational.
- Inverse ((¬ p) → (¬ q)): If π is not irrational, then 3 is not less than 2.
- Contrapositive ((¬ q) → (¬ p)): If 3 is not less than 2, then π is not irrational.
If p is true and q is false, what are the truth values of p → q, (¬ q) → (¬ p), q → p, and (¬ p) → (¬ q)?
If p is true and q is false:
- p → q is false.
- (¬ q) → (¬ p) is false.
- q → p is true.
- (¬ p) → (¬ q) is true.
How is the negation of “π is irrational” written, and why?
The negation is written as “π is not irrational” to emphasize that the opposite of being irrational is not necessarily assumed to be rational, unless otherwise stated.
What is a biconditional statement p ↔ q?
A biconditional statement p ↔ q, read as “p if and only if q,” is true only if both p and q are true or both p and q are false.
What is the truth table for p ↔ q?
p | q | p ↔ q |
|— |— |——— |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
What is a tautology?
A tautology is a compound proposition whose truth value remains true regardless of the truth values of its component propositions.
What is a contradiction?
A contradiction is a compound proposition whose truth value remains false regardless of the truth values of its component propositions.
Give an example of a tautology and a contradiction.
The compound statement p ∨ (¬ p) is a tautology, while the compound statement p ∧ (¬ p) is a contradiction.
What is the truth table for p ∨ (¬ p) and p ∧ (¬ p)?
p | ¬ p | p ∨ (¬ p) | p ∧ (¬ p) |
|—|——|————–|————-|
| 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 |
What does it mean for p to logically imply q?
For p to logically imply q, expressed as p ⇒ q, means that the conditional statement p → q is a tautology.
What does it mean for p and q to be logically equivalent?
p and q are logically equivalent, written as p ⇐⇒ q, if both p ⇒ q and q ⇒ p are true.
What is a contingency?
A contingency is a compound proposition that is neither a tautology nor a contradiction.
Provide an example that demonstrates p ⇒ p ∨ q and p ∧ q ⇒ p using a truth table.
| (p ∧ q) → p |
|—————–|
| 1 |
| 1 |
| 1 |
| 1 |
p | q | p ∨ q | p ∧ q | p → (p ∨ q) |
|—|—-|———|———|——————|
| 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 |
What is the law of addition in logic?
The law of addition is the implication p ⇒ (p ∨ q). It states that if p is true, then p ∨ q (p or q) is also true.
What is the law of simplification in logic?
The law of simplification is the implication (p ∧ q) ⇒ p. It means that if both p and q are true, then p must be true.
