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contraposition

contraposition (ˌkɒntrəpəˈzɪʃən) [ad. L. contrāpositiōn-em (Boethius), n. of action from contrāpōnĕre to contrapone.] 1. A placing over against; antithesis, opposition, contrast. Phr. in contraposition to (or with).1581 J. Bell Haddon's Answ. Osor. 332 A figure called contraposition betwixt the decr... Oxford English Dictionary

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Contraposition

an associated proof method known as proof by contraposition. Hence, Bayes' theorem represents a generalization of contraposition. wikipedia.org

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Contraposition (traditional logic)

The schema of contraposition: Notice that contraposition is a valid form of immediate inference only when applied to "A" and "O" propositions. The contrapositive is the product of the method of contraposition, with different outcomes depending upon whether the contraposition is full, or partial wikipedia.org

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Validity of Contraposition. Contraposition says: $$ P \implies Q \iff \neg Q \implies \neg P $$ What if P means "Santa Claus is in town" and Q means "I am in town"? It would mean: If Santa Claus is in town, then I am...

Don't think of "cause" here - it doesn't generalize that easily. Consider the statements P = "The Aggies win the next football game" and Q = "I win the bet I made with Tom." So, $P \implies Q$ means "If the Aggies win the next football game, then I win the bet I made with Tom." Consider the contrapo...

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Contraposition of "P if and only if Q" I'm understanding the basic idea of contraposition, when it comes to propositional logic and writing proofs, but I'm having trouble figuring out what the contraposition of "P if ...

$P \Leftrightarrow Q$ is the same as $P \Rightarrow Q$ and $Q \Rightarrow P$. So contrapose those two and you get $Not(Q) \Rightarrow Not(P)$ and $Not(P) \Rightarrow Not(Q)$, otherwise written as $Not(P) \Leftrightarrow Not(Q)$

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Contraposition of an implication with quantifiers I am trying to prove a theorem, and a method is by using contraposition. What is the contraposition of the phrase: $\exists x$ satisfying P $\Rightarrow$ $\forall y$ ...

You are correct. $\forall$ turns into $\exists$ and each logical expression gets negated.

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Need Help Using Proof by Contraposition > Let $x$ be a real number. Prove that if $x^3+5x+1\le0$, then $x<0.$ My solution: Suppose that $x\ge0$, what should I do next?

So, by contraposition we have shown the statement.

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How is the law of contraposition a tautology? I recently started the study of Aristotelian logic in Math class. I wanted to ask (as the title suggests) why the law of contraposition is a tautology. My book states that...

Just fill in the truth table like this: P | Q | P → Q | ~Q → ~P | (P → Q) ↔ (~Q → ~P) ––|–––|–––––––|––––––––––|–––––––––––––––––––– T | T | T | T | T T | F | F | F | T F | T | T | T | T F | F | T | T | T As you see, these two have identical truth values, and they are _logically equivalent_. It mean...

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Using proof by contraposition to show that if $n\in\mathbb Z$ and $3n+2$ is even, then $n$ is even I have my answer below but there is one step that I am not understanding...and maybe my brain is just not trained to u...

Well, the contrapositive of ' _$A$ implies $B$_ ' is ' _not $B$ implies not $A$_ '. In this case, now -staying withing the realm of integers- it would read > If $n$ is odd, then $3n+2$ is odd. And this is being proved. The equality sign on the second line is not correct, as $n=2k+1\ne 3n+2$, so it s...

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Contraposition and law of excluded middle Does truth-equivalence of an $A \rightarrow B$ and contrapositive $\neg B \rightarrow \neg A$ rely on the law of excluded middle?

It would seem that it does not, but rather, on the law of non-contradiction. Assume 1. $A\implies B$ 2. $\neg B$ Now, to obtain a contradiction, suppose $A$. Then we have $B$ from (1). But we have $\neg B$ from (2), so by the law of non-contradiction, it follows that we have $\neg A$, and finally we...

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Domination and Contraposition Laws - Discrete Math Im having quite a bit of trouble understanding the Domination and Contraposition Laws in the instance below. I just do not see how the Domination Law, $\rho \wedge \m...

The 6th line is derived from the 4th and 5th line by the law of Contraposition: if something of the form $A\to B$ is proven and $\sim\\!

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How to prove universally-quantified formula is true by contraposition? > For all natural numbers $x$ and $y$, if $x+y$ is odd, then $x$ is odd or $y$ is odd. How do I prove the following statement is true by contrapo...

In that case the statement only works in one direction, and the proof by contraposition involves proving the statement "$x$ even and $y$ even implies $

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Proof with symmetric matrix (Direct and Contraposition) $S$ is a symmetric matrix. If $Sx=\lambda x$ and $Sy = \mu y$, where $x$ and $y$ are non-zero vectors and $\lambda , \mu \in \mathbb{R}$. Prove the following: (...

## Part a $$\begin{array}{rcl} \lambda x^T y &=& (x^T S^T) y \\\ &=& (x^T S) y \\\ &=& x^T (S y) \\\ &=& x^T (\mu y) \\\ \end{array}$$ Therefore $(\lambda-\mu)(x^T y) = 0$ ## Part b $$\begin{array}{rcl} S x &=& \lambda x \\\ S^2 x &=& \lambda^2 x \\\ S^3 x &=& \lambda^3 x \\\ S^4 x &=& \lambda^4 x \...

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Suppose $a,b∈\mathbb{R}$ use proof by contrapositive to show the following implication is true $b^3+ba^2\leq a^3+ab^2 \Rightarrow b\leq a$ Suppose $\\{a,b\\}\subset \mathbb{R}$ use proof by contraposition to show the ...

For $ab=0$ it's obvious. Let $ab\neq0$ and $b>a$. Thus, $$b^3+a^2b-a^3-ab^2=(b-a)(a^2+b^2)>0.$$ We got a contradiction.

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Prove by either direct proof or contraposition I have a question like this: By direct proof or by contraposition: Let $a \in Z$, if $a \equiv 1 \pmod{5}$, then $a^2 \equiv 1 \pmod{5}$. Hypothesis: $a \in Z,~a \equiv...

**Direct Proof:** Using the property of Modular arithmetic, If $x\equiv{c}\pmod{d}$, where $x,c,d\in{\Bbb{Z}}$, then $x^n\equiv{c^n}\pmod{d}$, where $n\in{\Bbb{Z}}$ Since $a\equiv{1}\pmod{5}$ So $a^2\equiv{1^2}\equiv1\pmod{5}$ **Non-modular Proof** Let $a=5k+1$, $k\in{\Bbb{N}}$ $a^2=(5k+1)^2=25k^2+1...

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contraposition

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contraposition

Contraposition

Contraposition (traditional logic)

Validity of Contraposition. Contraposition says: $$ P \implies Q \iff \neg Q \implies \neg P $$ What if P means "Santa Claus is in town" and Q means "I am in town"? It would mean: If Santa Claus is in town, then I am...

Contraposition of "P if and only if Q" I'm understanding the basic idea of contraposition, when it comes to propositional logic and writing proofs, but I'm having trouble figuring out what the contraposition of "P if ...

Contraposition of an implication with quantifiers I am trying to prove a theorem, and a method is by using contraposition. What is the contraposition of the phrase: $\exists x$ satisfying P $\Rightarrow$ $\forall y$ ...

Need Help Using Proof by Contraposition > Let $x$ be a real number. Prove that if $x^3+5x+1\le0$, then $x<0.$ My solution: Suppose that $x\ge0$, what should I do next?

How is the law of contraposition a tautology? I recently started the study of Aristotelian logic in Math class. I wanted to ask (as the title suggests) why the law of contraposition is a tautology. My book states that...

Using proof by contraposition to show that if $n\in\mathbb Z$ and $3n+2$ is even, then $n$ is even I have my answer below but there is one step that I am not understanding...and maybe my brain is just not trained to u...

Contraposition and law of excluded middle Does truth-equivalence of an $A \rightarrow B$ and contrapositive $\neg B \rightarrow \neg A$ rely on the law of excluded middle?

Domination and Contraposition Laws - Discrete Math Im having quite a bit of trouble understanding the Domination and Contraposition Laws in the instance below. I just do not see how the Domination Law, $\rho \wedge \m...

How to prove universally-quantified formula is true by contraposition? > For all natural numbers $x$ and $y$, if $x+y$ is odd, then $x$ is odd or $y$ is odd. How do I prove the following statement is true by contrapo...

Proof with symmetric matrix (Direct and Contraposition) $S$ is a symmetric matrix. If $Sx=\lambda x$ and $Sy = \mu y$, where $x$ and $y$ are non-zero vectors and $\lambda , \mu \in \mathbb{R}$. Prove the following: (...

Suppose $a,b∈\mathbb{R}$ use proof by contrapositive to show the following implication is true $b^3+ba^2\leq a^3+ab^2 \Rightarrow b\leq a$ Suppose $\\{a,b\\}\subset \mathbb{R}$ use proof by contraposition to show the ...

Prove by either direct proof or contraposition I have a question like this: By direct proof or by contraposition: Let $a \in Z$, if $a \equiv 1 \pmod{5}$, then $a^2 \equiv 1 \pmod{5}$. Hypothesis: $a \in Z,~a \equiv...