Can a mathematical statement be both true and false?

In mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition. The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both.

Why is implication true when p is false?

If p is true then q. So when p is true then the value of q is the value of the whole statement. When p is false the implication doesn’t say anything about q, that is no ‘prediction’ is being made and thus the statement hasn’t been falsified. So if p is false then p⟹q is always true.

How do you know if a mathematical expression is true or false?

A number sentence is a statement of equality between two numerical expressions. A number sentence is said to be true if both numerical expressions are equivalent (that is, both evaluate to the same number). It is said to be false otherwise.

What is a statement that is either true or false?

Proposition
Proposition is simply a statement that is either true or false, has no variables involved.

Is false and false true?

false and false is false logically.

What does true/false mean?

true-false – offering a series of statements each of which is to be judged as true or false; “a true-false test” multiple-choice – offering several alternative answers from which the correct one is to be chosen; or consisting of such questions; “multiple-choice questions”; “a multiple-choice test”

Is a declarative sentence that can be meaningfully classified as either true or false?

A proposition is a declarative sentence that is either true or false (but not both).

Are P → Q and P ∨ Q logically equivalent?

(p → q) and (q ∨ ¬p) are logically equivalent. So (p → q) ↔ (q ∨ ¬p) is a tautology. Thus: (p → q)≡ (q ∨ ¬p).

Why True implies false is false?

The reason false imply true is true is that Q can be true by itself even if P is false. if win then celebrate. But u can celebrate even if win is false; the statement is still hold valid. The reason true imply false is false is that this statement is invalid.

Why is the compound proposition p ∧ q therefore false?

The compound proposition p ∧ q is therefore false, because it is not the case that both propositions are true. A truth table shows the truth value of a compound proposition for every possible combination of truth values for the variables contained in the compound proposition.

What is the logical equivalent of 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.”

When are two statements logically equivalent?

Two (molecular) statements P and Q are logically equivalent provided P is true precisely when Q is true. That is, P and Q have the same truth value under any assignment of truth values to their atomic parts.

Is the implication of a hypothesis always true?

Still, it is important to remember that an implication is a statement, and therefore is either true or false. The truth value of the implication is determined by the truth values of its two parts. To agree with the usage above, we say that an implication is true either when the hypothesis is false, or when the conclusion is true.