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
P
→
Q
{\displaystyle P\rightarrow Q}
is
¬
Q
→
¬
P
{\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
P
→
Q
{\displaystyle P\rightarrow Q}
:
Inversion (the inverse),
¬
P
→
¬
Q
{\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),
Q
→
P
{\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).
Negation,
¬
(
P
→
Q
)
{\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
P
→
Q
{\displaystyle P\rightarrow Q}
is true and one is given that
Q
{\displaystyle Q}
is false (i.e.,
¬
Q
{\displaystyle \neg Q}
), then it can logically be concluded that
P
{\displaystyle P}
must be also false (i.e.,
¬
P
{\displaystyle \neg P}
). This is often called the law of contrapositive, or the modus tollens rule of inference.
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
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) ##...
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...
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...
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...
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...
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...
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...
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?
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...
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...
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
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
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...
"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...
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...
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
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...
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...
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...
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...
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)...
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...
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...
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)
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...
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...
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...
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...
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...
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...
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...
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...
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...
"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...