How can we reason about “if P then Q” or “P only if Q” statements in propositional logic?

The statement ‘if P, then Q’ means that whenever P is true, Q must also be true. The only way for ‘if P, then Q’ to not be true is if it is false. For the statement ‘if P, then Q’ to be false, we need to have P be true, but Q be false.

What do you call a statement P if and only if q?

In conditional statements, “If p then q” is denoted symbolically by “p q”; p is called the hypothesis and q is called the conclusion. For instance, consider the two following statements: If Sally passes the exam, then she will get the job. If 144 is divisible by 12, 144 is divisible by 3.

What proposition can be shown to be logically equivalent to if/p then q by a truth table?

The propositions are equal or logically equivalent if they always have the same truth value. That is, p and q are logically equivalent if p is true whenever q is true, and vice versa, and if p is false whenever q is false, and vice versa.

Logical Arguments as Compound Propositions.

p q [(p → !q)& p ] → !q

Is only if and if and only if the same?

IF AND ONLY IF, is a biconditional statement, meaning that either both statements are true or both are false. So it is essentially and “IF” statement that works both ways. Note that IF AND ONLY IF is different than simply ONLY IF.

How do you write if/p then q statements?

If P then Q you want to say that there is an initial statement I hypothesis P and that if that hypothesis is true then the conclusion the second statement Q must also be true.

How do you prove if and only if statements?

To prove a theorem of the form A IF AND ONLY IF B, you first prove IF A THEN B, then you prove IF B THEN A, and that’s enough to complete the proof.

Where p and q are statements p q is called the?

The statement p is called the hypothesis of the implication, and the statement q is called the conclusion of the implication. The biconditional or double implication p ↔ q (read: p if and only if q) is the statement which asserts that p and q if p is true, then q is true, and if q is true then p is true.

What is only if in propositional logic?

In logic and related fields such as mathematics and philosophy, “if and only if” (shortened as “iff”) is a biconditional logical connective between statements, where either both statements are true or both are false.

Why do we use if and only if?

It is often used to conjoin two statements which are logically equivalent. In general, given two statement A and B, the statement “A if and only if B” is true precisely when both A and B are true or both A and B are false. In which case, A can be thought of as the logical substitute of B (and vice versa).

When phrased as a conditional statement an argument is valid if and only if it is a?

Valid: an argument is valid if and only if it is necessary that if all of the premises are true, then the conclusion is true; if all the premises are true, then the conclusion must be true; it is impossible that all the premises are true and the conclusion is false.

How do you write a proof by contradiction?

How do you do proof by contradiction?

  1. Step 1: Take the statement, and assume that the contrary is true (i.e. assume the statement is false).
  2. Step 2: Start an argument, starting from the assumed statement, and try to work towards the conclusion.
  3. Step 3: While doing so, you should reach a contradiction.

What does converse mean in logic?

converse, in logic, the proposition resulting from an interchange of subject and predicate with each other. Thus, the converse of “No man is a pencil” is “No pencil is a man.” In traditional syllogistics, generally only E (universal negative) and I (particular affirmative) propositions yield a valid converse.

How do you prove a contradiction?

To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequences of this are not possible. That is, the consequences contradict either what we have just assumed, or something we already know to be true (or, indeed, both) – we call this a contradiction.

What do you understand by contradiction in propositional calculus?

; a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition). This can be generalized to a collection of propositions, which is then said to “contain” a contradiction.

What is a logical contradiction?

A logical contradiction is the conjunction of a statement S and its denial not-S. In logic, it is a fundamental law- the law of non contradiction- that a statement and its denial cannot both be true at the same time. Here are some simple examples of contradictions. 1. I love you and I don’t love you.

Why do we use proof by contradiction?

Proving that Something Does Not Exist

In mathematics, we sometimes need to prove that something does not exist or that something is not possible. Instead of trying to construct a direct proof, it is sometimes easier to use a proof by contradiction so that we can assume that the something exists.

What is the difference between proof by contradiction and proof by Contrapositive?

In a proof by contrapositive, we actually use a direct proof to prove the contrapositive of the original implication. In a proof by contradiction, we start with the supposition that the implication is false, and use this assumption to derive a contradiction. This would prove that the implication must be true.

How do you prove a statement is false?

A counterexample disproves a statement by giving a situation where the statement is false; in proof by contradiction, you prove a statement by assuming its negation and obtaining a contradiction.

How do you use logical reasoning to prove statements are true?

The trick to using logical reasoning is to be able to support any statement (conjecture) you make with a valid reason. In geometry, we use facts, postulates, theorems, and definitions to support conjectures. Watch the video to see what can go wrong if you don’t support your conjecture.

How do you prove a statement is true?

There are three ways to prove a statement of form “If A, then B.” They are called direct proof, contra- positive proof and proof by contradiction. DIRECT PROOF. To prove that the statement “If A, then B” is true by means of direct proof, begin by assuming A is true and use this information to deduce that B is true.

How do you prove a conditional statement?

There is another method that’s used to prove a conditional statement true; it uses the contrapositive of the original statement. The contrapositive of the statement “If (H), then (C)” is the statement “If (the opposite C), then (the opposite of H).” We sometimes write “not H” for “the opposite of H.”

How do you prove for all statements?

Write our plus s as a over B and then we can prove that these two things both can't be true remember if this is false then at least one of them has to be false.

How can you determine the hypothesis and the conclusion in each statement?

A conditional statement (also called an if-then statement) is a statement with a hypothesis followed by a conclusion. The hypothesis is the first, or “if,” part of a conditional statement. The conclusion is the second, or “then,” part of a conditional statement. The conclusion is the result of a hypothesis.