What is Contrapositive: Definition and 36 Discussions

In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped. For instance, the contrapositive of the conditional statement "If it is raining, then I wear my coat" is the statement "If I don't wear my coat, then it isn't raining." In formulas: the contrapositive of



{\displaystyle P\rightarrow Q}



{\displaystyle \neg Q\rightarrow \neg P}
. The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true.The contrapositive can be compared with three other conditional statements related to



{\displaystyle P\rightarrow Q}

Inversion (the inverse),



{\displaystyle \neg P\rightarrow \neg Q}

"If it is not raining, then I don't wear my coat." Unlike the contrapositive, the inverse's truth value is not at all dependent on whether or not the original proposition was true, as evidenced here.
Conversion (the converse),



{\displaystyle Q\rightarrow P}

"If I wear my coat, then it is raining." The converse is actually the contrapositive of the inverse, and so always has the same truth value as the inverse (which as stated earlier does not always share the same truth value as that of the original proposition).



{\displaystyle \neg (P\rightarrow Q)}

"It is not the case that if it is raining then I wear my coat.", or equivalently, "Sometimes, when it is raining, I don't wear my coat. " If the negation is true, then the original proposition (and by extension the contrapositive) is false.Note that if



{\displaystyle P\rightarrow Q}
is true and one is given that


{\displaystyle Q}
is false (i.e.,


{\displaystyle \neg Q}
), then it can logically be concluded that


{\displaystyle P}
must be also false (i.e.,


{\displaystyle \neg P}
). This is often called the law of contrapositive, or the modus tollens rule of inference.

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  1. docnet

    Write the contrapositive and negation of the statement

    The contrapositive statement is $$\forall x\in\mathbb{R}, (x^2\leq 5)\Rightarrow (x<3)$$ The negation statement is $$\forall x\in\mathbb{R}, (x<3)\Rightarrow (x^2\leq 5)$$ o_O
  2. C

    Changing the Statement Combinatorial proofs & Contraposition

    I have a question regarding to combinatorial proofs and predicate logic. It seems to me that in some combinatorial proofs there is a use of contraposition ( although not explicitly stated in the books where I've read so far ), for example If we to prove that ## C(n,k) = C(n,n-k) ##...
  3. V

    Finding Converse and Contrapositive

    Homework Statement Find the converse and contrapositive of the statement: If n2 is even, then n is even. Homework EquationsThe Attempt at a Solution Converse: If n is even, then n2 is even. Contrapositive: If n is not even, then n2 is not even. Can someone check these over for me to make...
  4. S

    MHB Converse, Contrapositive and Negation for multiple Quantifiers

    If every printer is busy then there is a job in the queue. where B(p) = Printer p is busy and Q(j) = Print job j is queued. When it's translated to symbol, we'll have (∀pB(p)) → (∃jQ(j)). I'm trying to translate this statement to both English and symbol forms for Converse, Contrapositive and...
  5. Mr Davis 97

    I Contrapositive of quantified statement

    I have the following statement: Let ##a,b \in \mathbb{R}##. If ##a \le b_1##, for every ##b_1 > b##, then ##a \le b##. I have put it into logical notation in the following way: ##\forall a,b, b_1 \in \mathbb{R} ((b_1 > b \rightarrow a \le b_1) \rightarrow a \le b)##. My question is, if I want to...
  6. J

    I Can you make the induction step by contradiction?

    Assuming you've sufficiently proven your inductive basis, can you complete a proof by induction in the following manner: Make the inductive hypothesis, assume P(n) is true for some n. Assume P(n+1) is not true. If it follows from the assumption that P(n+1) is false that P(n) must also...
  7. Y

    Proofs involving Negations and Conditionals

    Suppose that A\B is disjoint from C and x∈ A . Prove that if x ∈ C then x ∈ B . So I know that A\B∩C = ∅ which means A\B and C don't share any elements. But I don't necessarily understand how to prove this. I heard I could use a contrapositive to solve it, but how do I set it up. Which is P...
  8. kmas55

    Prove Using the Method of Contrapositive

    Prove both by method of contrapositive. 1. If a ≤ b + ε, where ε > 0, then b > a. 2. If 0 ≤ a - b < ε, where ε > 0, then a = b. I'll start with problem 1.: p: If a ≤ b + ε, where ε > 0 q: b > a neg q: b < a neg p: for some ε' > 0 1/2(a - b), a > b + ε' define ε' = 1/2(a - b) I...
  9. T

    MHB Contrapositive Proof: Ints $m$ & $n$ - Even/Odd Combinations

    For all integers $m$ and $n$, if $m+ n$ is even then $m$ and $n$ are both even or both odd. For a contrapositive proof, I need to show that for all ints $m$ and $n$ if $m$ and $n$ and not both even and not both odd, then $ m + n $ is not even. How do I go about doing this?
  10. J

    Proofs involving negations and conditionals

    0. Background First and foremost, this is a proof-reading request. I'm going through Velleman's "How To Prove It" because I found that writing and understanding proofs is a prerequisite to serious study of mathematics that I did not meet. Unfortunately, the book is very light on answers to its...
  11. C

    MHB Can L-Shaped Tiles Fit Perfectly on a 2xn Board If n Is Not Divisible by 3?

    Say there's a 3 block/pixel/square shape in L- formation that can be rotated on a board of size 2 x n. G(n) is how many distinct ways the board can be tiled. I need to show that if n isn't divisible by 3, then G(n) is 0. Given a block of three squares fitting on a board of size 2xn, and k...
  12. A

    MHB Converse,the contrapositive and the inverse of these condition

    q)give the converse ,the contrapositive and inverse of these conditional statements a)if it rains today,then i will drive to work b)if |x|=x then x>=0 c)if n is greater than 3,then n^2 is greater then 9
  13. U

    Proving with contrapositive methode instead of contradition

    Could you tell me how to write a very formal proof of the statement below with the contrapositive methode, if possible. (I know how to do it with contradiction) Let x be a rational number and y an irrational number, then x times y is irrational. V. Uljanov
  14. J

    MHB Write inverse, converse, and contrapositive following statement

    Write the inverse, converse, and contrapositive of the following statement: upside down A x E R, if (x + 2) (x - 3) > 0, then x < -2 or x > 3 Indicate which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a conterexample for each that...
  15. P

    Converse, inverse and contrapositive

    "All engineers have practical skills or are good at mathematics" to write down the converse, inverse and contrapositive for the above statement, I have to find the hypothesis and the conclusion of the statement. but how? is there any other way to write converse, inverse and contrapositive...
  16. B

    Taking the contrapositive of this statement?

    Statement: If every right triangle has angle defect equal to zero then the angle defect of every triangle is equal to zero Taking the contrapositive do i have this correct? : There exists at least one triangle whose angle defect is not zero such that not every right triangle has an angle defect...
  17. B

    Please check my Contrapositive statement

    I am trying to write a CP for: Every connected M. Space with at least 2 points is uncountable. Restatement: if a MS X is connected with |X|≥ 2 => X is uncountable. Contrapositive: a MS X has only one point => X is not connected. Thanks
  18. S

    Contrapositive of a (if p and q, then r) statement?

    Homework Statement I hope this is the right place to post this. For my linear algebra homework, I have to prove that "If \vec{u} \neq \vec{0} and a\vec{u} = b\vec{u}, then a = b." Homework Equations The Attempt at a Solution I'm trying to prove the contrapositive, but I'm not sure...
  19. B

    What would be the contrapositive of this statement?

    Homework Statement The original statement is Prove that if xy and x+y are even then both x and y are even. Homework Equations The Attempt at a Solution I think it goes like "If x or y is odd then xy and x+y are odd"? I'm not too sure though because the first "and" in the...
  20. J

    Proof by contrapositive; if (m^2+n^2) div by 4, then m,n are even numbers

    Homework Statement Let m and n be two integers. Prove that if m^2 + n^2 is divisible by 4, then both m and n are even numbers Homework Equations The Attempt at a Solution Proof by Contrapositive. Assume m, n are odd numbers, showing that m^2 + n^2 is not divisible by 4...
  21. T

    Contrapositive Proof of Theorem: x > y → x > y+ε

    Homework Statement Theorem: Let x,y,ε be ℝ. If x≤ y+ε  for every ε > 0 then x ≤ y. Write the above as a logic statement and prove it using contrapositive proof. The attempt at a solution The contrapositive statement x > y → x > y+ε is only true if ε < 0. Does a...
  22. F

    Contrapositive proof of irrational relations

    I'm confused with a question and wondered if anyone could help explain where I need to go... let x ε R. Prove that x is irrational thenI'm confused with a question and wondered if anyone could help explain where I need to go... let x ε R. Prove that x is irrational then ((5*x^(1/3))-2)/7)...
  23. D

    Is the Contrapositive Law Demonstrably Useful in Set Theory Proofs?

    I ran into some difficulties trying to show "prove" the contrapositive law (CPL). I remember in first year my professor showed that P ⇒ Q is logically equivalent to ¬Q ⇒ ¬P by showing that the truth tables for both statements were the same for all possible truth values of P and Q. Statement...
  24. S

    Contrapositive: Basic but Makes No Sense | xy=0, x!=0 & y!=0

    This is supposedly basic but it makes no sense to me. The other topic was very old so I decided to just start a new one. Given: If x=0 and y=0 then xy=0. They say the contrapositive(which they say is always true) is: If xy!=0 then x!=0 OR y!=0. But that is exactly false, because...
  25. A

    Proofs using contrapositive or contradiction

    Please Help! proofs using contrapositive or contradiction Homework Statement Prove using contrapositive or contradiction: For all r,s∈R,if r and s are positive,then √r+ √s≠ √(r+s)
  26. R

    Set Theory - Proving Contrapositive

    Homework Statement using set theroetic notation, write down and prove the contra-positive of: GOD WHAT IS WRONG WITH LATEX? It is completely ruining my set notation! And i can't fix it! If B \cap C \subseteq A Then (C-A) u (B-A) is empty. The Attempt at a Solution I'm awful with set...
  27. R

    Proof by contrapositive = modus tollens?

    I was just looking at the http://en.wikipedia.org/wiki/Modus_tollens" and found the line "Modus tollens is sometimes confused with proof by contradiction or proof by contrapositive." I thought proof by contrapositive and modus tollens are one and the same though. Is that then not the case or is...
  28. 2

    Have I Understood the Process of Proof by Contrapositive Correctly?

    Hi guys, I've just started university this week and I've been given a mountain of assignments. One of them has a proof question in it. Since this is an assignment I want to make clear that I don't want help with the actual proof. In the first part of the question I'm asked to, given a...
  29. T

    Help, Negation and Contrapositive of the following statement?

    Homework Statement Consider the statement "if x is odd and x is a multiple of 3, then x ≥ 6." write down the contrapositive and negation of this statement? 2. The attempt at a solution This is what I worked our as an answer but I am pretty sure it's wrong. Contrapositive "if x ≤ 6, then x...
  30. S

    Is this the correct contrapositive?

    Hello, I have a real simple question. Given, If x and y are two integers whose product is even, then at least one of the two must be even. Is the contrapositive, If both x and y are odd, then the product of x and y is odd? Similarly, If x and y are two integers whose product is odd...
  31. K

    What is the Contrapositive Statement of Two Non-Negative Numbers?

    Fact: If a and b are non-negative numbers, then ab is non-negative. What is the equivalent contrapositive statement of the above? I think it is: If ab<0, then at least one of a and b <0. Am I right? But this implication doesn't seem quite right to me...shouldn't the correct statement...
  32. T

    Is Proving the Contrapositive Equivalent to Proof by Contradiction?

    It can be proved by proof by contradiction. hence it is just a variant of it?
  33. G

    Does the contrapositive statement require changing and to or?

    The statement is: If α is one-to-one and β is onto, then βoα is one-to-one and onto. One-to-one is injection, onto is surjection, and being both is bijection. After showing that the statement is false, the contrapositive was asked for. The answer given is: If βoα is not one-to-one and onto...
  34. G

    Contrapositive Proof of Positive x & y: x^n<y^n implies x<y

    Homework Statement 1st part , using induction to prvoe that if both x and y are positive then x<y implies x^n<y^n 2nd part, prove the converse, that if both x and y are postive then x^n<y^n implies x<y Homework Equations my question is more on the second part. I understand that I have to...
  35. mattmns

    Lin Alg - Matrix multiplication (Proof by contrapositive)

    Hello, here is the question my book is asking: Let A, B be two m x n matricies. Assume that AX = BX for all n-tuples X. Show that A = B. ------- So I decided to try and prove the contrapositive, which is (unless I am mistaken): If A \neq B, then there is some X such that AX \neq BX Proof...
  36. I

    Can a Non-Converging Bounded Sequence Have Subsequences with Distinct Limits?

    "If X is a bounded sequence that does not converge, prove that there exists at least two subsequences of X that converge to two distinct limits." There is a what I like to call "mass produced" version of the proof with limsup and liminf (which actually tells you where the two subsequences...