In most logical systems, one proves a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly.

2015年6月30日 · Here's a simpler way to do it: Use the fact that p ⇔ q ≡ (p ⇒ q) ∧ (¬p ⇒ ¬q) p ⇔ q ≡ (p ⇒ q) ∧ (¬ p ⇒ ¬ q) So, if x x is real, then f(x) = 1 f (x) = 1. If x x is not real, then f(x) = 0 f (x) = 0, i.e. f(x) ≠ 1 f (x) ≠ 1. This proves the statement. You must log in to answer this question. Find the answer to your question by asking.

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. Using this technique, you can use IF...THEN proofs as well as IF AND ONLY IF proofs in your own proof.

Suppose I am trying to prove a statement in the form A if and only if B. I know I need to prove that. I know that 1 is equivalent to proving "If not B, then not A". My question is: When proving A if and only if B, is it permissible to prove "if not B, then not A" and then "if B, then A."

Proof 7.2 Equivalent Statements 7.3 Existence and Uniqueness Proofs 7.4 (Non-) Construc-tive Proofs Lists of Equivalent Statements Theorem: Suppose A is an n n matrix. The following statements are equivalent: (a)The matrix A is invertible. (b)The equation Ax = b has a unique solution for every b 2Rn.

2021年6月30日 · The statement “\(P \text{ IFF } Q\)” is equivalent to the two statements “\(P \text{ IMPLIES } Q\)” and “\(Q \text{ IMPLIES } P\).” So you can prove an “iff” by proving two implications: Write, “We prove \(P\) implies \(Q\) and vice-versa.” Write, “First, we …

3、 Proof: 就是证明的开始，告诉别人我开始写具体的证明过程了. 4、 s.t. 使得…满足…的意思，是subject to 的缩写,其实也可以是such that. 举个例子： \forall x s.t.x>0 ，意思就是任意一个 x 都满足大于 0. 5、 LHS,RHS. 其实就是Left Hand Side（左边 = …），Right Hand Side ...

2019年5月22日 · If you assumed that B B is untrue, and proved that A A is then untrue, you have proven the statement ¬B ¬A ¬ B ¬ A. This statement is logically equivalent to the statement A B A B, which means you proved the direction from left to right. You still have to prove either B A B A or its equivalent, ¬A ¬B ¬ A ¬ B.

I’m writing this short note as a reminder to myself what is meant by an ‘if and only’ (iff) proof. When I read a theorem that claims A iffB, (1) where A and B are propositions, I always have to figure out from first principles what this seemingly simple statement means. Well, what does the ‘iff’ in (1) mean? It means the conjunction ...

2025年1月31日 · To prove “if and only if” (P if and only if Q): Prove both the direct implication (P → Q) and the converse (Q → P). Prove the contrapositive (¬P → ¬Q). Prove: A number is prime if and only if it has only two factors, 1 and itself. – Assume A is prime. – By definition, a prime number has only two factors: 1 and itself.