How is if/p then q equivalent to not P or Q?
p only if q means “if not q then not p, ” or equivalently, “if p then q.” Biconditional (iff): The biconditional of p and q is “p if, and only if, q” and is denoted p q. It is true if both p and q have the same truth values and is false if p and q have opposite truth values.
How do you know if a statement is logically equivalent?
Logical equivalence occurs when two statements have the same truth value. This means that one statement can be true in its own context, and the second statement can also be true in its own context, they just both have to have the same meaning.
Why is p implies q true when p is false?
The implication p → q (read: p implies q, or if p then q) is the state- ment which asserts that if p is true, then q is also true. We agree that p → q is true when p is false. The statement p is called the hypothesis of the implication, and the statement q is called the conclusion of the implication.
How do you determine if a conditional statement is true or false?
A conditional is considered true when the antecedent and consequent are both true or if the antecedent is false. When the antecedent is false, the truth value of the consequent does not matter; the conditional will always be true.
Is P → Q → [( P → Q → Q a tautology Why or why not?
A compound proposition that is always True is called a tautology. Two propositions p and q are logically equivalent if their truth tables are the same. Namely, p and q are logically equivalent if p ↔ q is a tautology. If p and q are logically equivalent, we write p ≡ q.
What is equivalent to P → Q?
P → Q is logically equivalent to ¬ P ∨ Q . Example: “If a number is a multiple of 4, then it is even” is equivalent to, “a number is not a multiple of 4 or (else) it is even.”
What is equivalent to p implies q?
Thus, “p implies q” is equivalent to “q or not p”, which is typically written as “not p or q”. This is one of those things you might have to think about a bit for it to make sense, but even with that, the truth table shows that the two statements are equivalent.
What can you conclude about P and Q If you know the statement is true?
Make a truth table for the statement ¬P∧(Q→P). What can you conclude about P and Q if you know the statement is true? If the statement is true, then both P and Q are false.
What does P only if Q mean?
Only if introduces a necessary condition: P only if Q means that the truth of Q is necessary, or required, in order for P to be true. That is, P only if Q rules out just one possibility: that P is true and Q is false.
Why is if false then true true?
5 Answers. As an example of why the convention ‘false implies true is true’ is useful, consider the sentence “if a given number is smaller than 10 then it is also smaller than 100”. This is clearly a true statement. This is an example of ‘false implies true’, and it still should be a true statement.
Which of the following statement makes an IF THEN statement false?
Note that a conditional statement is only false when the hypothesis is true and the conclusion is false. Also note that any conditional statement with a false hypothesis is trivially true….If-Then Statements.
| \begin{align*}P\end{align*} | \begin{align*}Q\end{align*} | \begin{align*}P \rightarrow Q\end{align*} |
|---|---|---|
| F | F | T |