Rd Sharma Xi 2020 2021 _volume 2 for Class 11 Science Maths Chapter 31 - Mathematical Reasoning

Answer:

​(i) The component statements of the given compound statement are:
1) The sky is blue.
2)The grass is green

(ii) The component statements of the given compound statement are:
1)The earth is round.
2)The sun is cold.

(iii) The component statements of the given compound statement are:
1) All rational numbers are real.
2) All real numbers are complex.

(iv) The component statements of the given compound statement are:
1) 25 is a multiple of 5.
2) 25 is a multiple of 8.

Answer:

​(i) Exclusive OR is used because students can opt for either Hindi or Sanskrit as their third language.

(ii) Inclusive OR is used because a person can have both passport as well as voter registration card.

(iii) Exclusive OR because a lady can give a birth to a baby who is either a boy or a girl.

(iv) Inclusive OR because a person could have both ration card as well as passport.

Answer:

​(i) The component statements of the given compound statement are:
1) To enter into a public library, children need an identity card from the school.
2) To enter into a public library, children need a letter from the school authorities.
The compound statement is true because both component statements are true.

(ii) The component statements of the given compound statement are:
1) All rational numbers are real.
2) All real numbers are not complex.
The compound statement is false because all real numbers are complex. The connective used is "and". So, even if one component statement is false, the compound statement is false.

(iii) The component statements of the given compound statement are:
1) Square of an integer is positive.
2) Square of an integer is negative.
The compound statement is true because the first statement is true. Since the connective used is "or" and one of the component statements is true, the compound statement is true.

(iv) The component statements of the given compound statement are:
1) $x=2$ is the root or the equation $3x^2-x-10=0$.
2) $x=3$ is the root or the equation  $3x^2-x-10=0$.
The connective used is "and". So, both component statements must be true for the compound statement to be true. The statement "$x=3$ is the root or the equation $3x^2-x-10=0$"  is false. Therefore, the compound statement is false.

(v) The component statements of the given compound statement are:
1) The sand heats up quickly in the sun.
2) Sand does not cool down fast at night.
The compound statement uses "and" as the connective. For the compound statement to be true, both the component statements must be true. The second component statement "Sand does not cool down fast at night" is false. Sand cools down fast at night. Therefore, the compound statement is false.

Answer:

(i) True
Both component statements are true and the connective is "and".
(ii) False
The first component statement "Delhi is in England" is false. Since the connective is "and" and one component statement is false, the compound statement is false.
(iii) False
The second component statement "2 plus 2 equals 5" is false. Since the connective is "and" and one component statement is false, the compound statement is false.
(iv) False
Both the component statements are false. Therefore, the compound statement is false.

Answer:

(i) Negation of the given statement:
There exists a number x such that $x+3\ge 10$.

(ii) Negation of the given statement:
For every x ϵ N, $x+3\ne 10$.

Answer:

(i) Negation of the given statement:
Some students did not complete their homework.
                                  
(ii) Negation of the given statement:
There exists a number which is not equal to its square.

                       

Answer:

(i) If you pay a subscription fee, then you can access the website.
(ii) If it rains, then there is a traffic jam.
(iii) If you want to log on to the server, then you need a passport.
(iv) If you want to be happy, then you will have to be rich.
(v) If it rains, only then the game is cancelled.
(vi) If it rains, then it is cold.
(vii) If it rains, then it is cold.
(viii) If it is cold, then it never rains.

Answer:

(i) Converse of the given statement:
If you feel thirsty, then it is hot outside.
Contrapositive of the given statement:
If you do not feel thirsty, then it is not hot outside.

(ii) Converse of the given statement:
If I go to a beach, then it is a sunny day.
Contrapositive of the given statement:
If I do not go to a beach, then it is not a sunny day.

(iii) Converse of the given statement:
If a positive integer has no divisors other than 1 and itself, then it is prime.
Contrapositive of the given statement:
If a positive integer has some divisors other than 1 and itself, then it is not prime.

(iv) Converse of the given statement:
If you have winter clothes, then you live in Delhi.
Contrapositive of the given statement:
​If you do not have winter clothes, then you do not live in Delhi.

(v) Converse of the given statement:
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Contrapositive of the given statement:
If the diagonals of a quadrilateral do not bisect each other, then it is not a parallelogram.

Answer:

(i) You watch television if and only if your mind is free.

(ii) A quadrilateral is a rectangle if and only if it is equiangular.

(iii) You get an A grade if and only if you do all the homework regularly.

(iv) The tumbler is half empty if and only if the tumbler is half full.

Answer:

(i) If Mohan is not poor, then he is not a poet.
(ii) If Max does not study, then he will not pass the test.
(iii) If she does not earn money, then she will not work.
(iv) If they do not drive the car, then there is no snow.
(v)  If it rains, it is not cold.
(vi) If it did not snow, then Ravish does not ski.
(vii) If x is positive, then x is not less than zero.
(viii) If he does not win, then he does not have courage.
(ix)  If you are not strong, then you cannot be a sailor.
(x)  If he tires, then he will not win.
(xi) If x is even, then x² is even.

Answer:

(i) p: 100 is a multiple of 4 and 5.
Since 100 is a multiple of 4 and 5, the statement is true.
Hence, it is a valid statement.

(ii) q: 125 is a multiple of 5 and 7.
Since 125 is a multiple of 5 but is not a multiple of 7, the statement is not true.
Hence, it is not a valid statement.

(iii) r: 60 is a multiple of 3 or 5.
Since 60 is a multiple of 3 and 5, the statement is true.
Hence, it is a valid statement.

Answer:

(i) p: If x and y are odd integers, then x + y is an even integer.
Let q and r be two statements.
Here,
q: x and y are odd integers.
r: x + y is an even integer.
Let q be true.
Then, q is true.
Now,
x and y are odd integers.
x = 2m +1 and y = 2n + 1 for some integers m and n.
$\Rightarrow$x + y = (2m + 1) + (2n + 1)
$\Rightarrow$x + y = 2m + 2n + 2 = 2(m + n + 1)
So, x + y is an even integer.
Hence, the statement is true.

ii) p: If x and y are integers such that xy is even, then at least one of x and y is an even integer.
Let q and r be two statements.
Here,
q: xy is an even integer.
r: At least one of x and y is an even integer.
Let r be not true.
Then, r is not true.
It is false that at least one of x and y is an even integer.
Now,
x and y are odd integers.
x = 2m +1 and y = 2n + 1 for some integers m and n.
$\Rightarrow$xy =(2m + 1)(2n + 1)
$\Rightarrow$xy = 2(2mn + m + n) + 1
So, xy is not an even integer. Thus, xy is not true.
Hence, the statement is true.

Answer:

p : "If x is a real number such that$x^{3 }+ x = 0$then x is 0".
Let q and r be the statements.
Here,
q: x is a real number such that $x^{3 }+ x = 0$.

r: x is 0.

(i) Direct method
Let q be true.
To obtain $x^{3 }+ x = 0$we have:
^{$x(x^2+1) = 0$}

or, x = 0
Thus, r is true.
Hence, "if q, then r" is a true statement.

(ii) Method of contrapositive
Let r not be true.
r is not 0.

If $x(x^2+1) \ne 0$, then q is not true.
Hence, "if ~q, then ~r" is a true statement.

(iii) Method of contradiction
Let q not be true.
Then,
$~$q is true
$~$(q $\Rightarrow$r) is true.

q &$~$r  is true


x is a real number such that $x^3+x=0$

Then, x is not 0.

x = 0 and x$\ne$0
This is a contradiction.
Hence, q is true.

Answer:

Let q and r be statements.
Here,
q: If x is an integer and x² is odd.
r: x is also odd.
Then, p is" if q, then r".
Let r be false.
So, x is not an odd integer, i.e., x is an even integer.
Let:
$x = 2n$ for some integer n
or, $x^2=4n^2$
In other words, x² is an even integer.
Now, q is false.
Thus, r is false; this implies q is false.
Hence, p is a true statement.

Answer:

The given statement can be rewritten as:
"The necessary and sufficient condition for integer n to be even is n² must be even".

Let p and q be the following statements.
p: The integer n is even.
q: n^​2 is even.
The given statement is "p if and only if q".
​To check its validity, we have to check the validity of the following statements:
(i) If p, then q.
(ii) If q, then p.
Checking the validity of "if p, then q"
"If the integer n is even, then n^​2 is even."
Let us assume that n is even.
Then, $n=2m$, where m is an integer.
Thus, we have:
$n^2=4m^2$
Here, n² is even.
Therefore, "if p, then q" is true.
The statement "if q, then p" is given by
"If n is an integer and n² is even, then n is even".
To check he validity of the statement, we will use the contrapositive method. So, let n be an integer. Then,
n is odd.
Here, $n=2k+1$ for some integer k.
$\Rightarrow$​$n^2=4k^2+2k+1$
Then, n^​2 is an odd integer.
n^​2 is not an even integer.
Thus "if q, then p" and "p if and only if q" are true.

Answer:

The given statement is of the form "if q, then r".

q: All the angles of a triangle are equal.

r: The triangle is an obtuse-angled triangle.

Statement p has to be proved false.
For this purpose, we need to prove that if q, then ~r.

To show this, none of the angles of the triangle should be obtuse.

We know that the sum of all angles of a triangle is 180°. Therefore, if all three angles are equal, then each of them will measure 60°, which is not an obtuse angle.

In an equilateral triangle, the measure of all angles is equal. Thus, the triangle is not an obtuse-angled triangle.

Hence, it can be concluded that statement p is false.

Answer:

(i) The given statement is false.

According to the definition of a chord, it should intersect the circumference of a circle at two distinct points.

(ii) The given statement is false.

If a chord is not the diameter of a circle, then the centre does not bisect that chord. In other words, the centre of a circle only bisects the diameter, which is the chord of the circle.

(iii) Equation of an ellipse:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

If we put a = b = 1, then we obtain $x^2+y^2=1$, which is an equation of a circle. Therefore, a circle is a particular case of an ellipse.

Thus, the statement is true.

(iv) x > y

⇒ –x < –y     (By the rule of inequality)

Thus, the given statement is true.

(v) 11 is a prime number and we know that the square root of any prime number is an irrational number. Therefore, $\sqrt{11}$ is an irrational number.

Thus, the given statement is false.

Answer:

The argument used to check the validity of the given statement is false because it is given in the argument that x² = π ² is irrational and therefore x = π is irrational; however, according to the statement, x is rational.

Answer:

Not all roses are black 
i.e. Roses are black is a false statement 
Hence, Roses are black is a statement 

Hence, the correct answer is option A.

Answer:

For statement = It is raining and weather is cold. 
~ negative statement is 
It is not raining or weather is not cold. (since negation of and is or) 
                             
Hence, the correct answer is option C. 

Answer:

For the statement,
If Billu secure good marks, then he will get a bicycle. 
Let p and q be two statements given by 
p: Billu secure good marks. 
q: Billu will get a bicycle. 
Given p ⇒ q
for q ⇒ p,
i.e. convert statement will be 
If Billu will a get a bicycle, then he will secure good makes
 
Hence, the correct answer is option B.

Answer:

a) x is a real number; 
The sentence is not always true, it depends an value of x.
Hence, it is not a statement 

b) Switch off the fan;
This sentence is not specific, i.e. no information regarding which fan.
Hence, it is not a statement 

c) 6 is a natural number; is a true statement 

d) Let me go, 
This sentence is not specific, have not a statement.

Hence, the correct answer is option C.

Answer:

a) Smoking is injurious to health; 
This sentence is always True; 
i.e. It is a true statement. 

b) 2 + 2 = 4; 
i.e. It is a true sentence. 

c) 2 is the only prime number. 
This sentence is false because infinitely many other prime numbers exist.
i.e. It is a false statement 

d) Come here, 
This sentence is not specific. 
Hence, not a statement. 

Hence, the correct answer is option D.

Answer:

a) The angle opposite to equal sides of a rectangle are equal; 
This sentence is always True; 
i.e. It is a true statement. 

b) The moon is a satellite of the earth; 
which is a universal  truth
i.e. It is a true statement. 

c) May god bless you; 
This sentence is not specific about the person, 
Hence, it is not a statement.

d) Asia is a continent; 
which is a universal truth
i.e. it is a true statement. 

Hence, the correct answer is option C. 

Answer:

For given statement, 
2 + 7 > 9 or 2 + 7 < 9
or is connective i.e or is connecting the two statements 
2 + 7 > 9 and 2 + 7 < 9

Hence, the correct answer is option B. 

Answer:

If p then q 
Means p ⇒ q 
center positive from means not q then not p. 
i.e. ~q ⇒ ~p
i.e. ~q then ~p

Hence, the correct answer is option C.

Answer:

For given statement; 
''7 is greater than 8" i.e. 7 > 8
negation is 7 is not greater than 8 
i.e. 7 ⋡ 8
Hence, the correct answer is option B. 

Answer:

For given statement; 
" A circle is an ellipse" 
i.e. circle ⇒ ellipse
Negation is circle ⇒ ellipse 
i.e. A circle is not an ellipse.

Hence, the correct answer is option C. 

Answer:

For given statement; 
"101 is not a multiple of 3
i.e. 101 ≠ 3k; for any k.
negation is 101 is a multiple of 3
i.e. 101 = 3k ; for some k.

Hence, the correct answer is option A. 

Answer:

For given statement, 
p: "72 is divisible by 2 and 3" 
~p: 72 is not divisible by 2 or 72 is not divisible by 3.

Hence, the correct answer is option is A.

Answer:

For given statement, 
p: " Rajni or Rajesh lived in Bangalore" 
~p: Rajesh did not live in Bangalore and Rajni did not live in Bangalore. 

Hence, the correct answer is option C.

Answer:

For given statement; 
p: "Plants take in  CO₂ and give out O₂ ''.
~p: "Plants do not take CO₂ or do not give out O₂".

Hence, the correct answer is option B.

Answer:

(i) Listen to me, Ravi!
It is an exclamatory sentence. Therefore, it is not a statement.

(ii) Every set is a finite set.
It is a false assertive sentence because there are some sets that are infinite like the set of all real numbers. Therefore, it is a statement.

(iii) Two non-empty sets have always a non-empty intersection.
It is a false assertive sentence. Two non-empty sets with no common elements can have an empty intersection. Therefore, it is a statement.

(iv) The pussy cat is black.
It is a declarative sentence, which may be true or false but cannot be both at the same time, so it is a statement.

(v) Are all circles round?
It is an interrogative sentence, so it is not a statement.

(vi) All triangles have three sides.
It is a true declarative sentence because a figure that has three sides is a triangle. Thus, it is a true statement.

(vii) Every rhombus is a square.
It is not true that every rhombus is a square because some rhombi may have all angles other than 90. So, it is a false statement.

(viii) x² + 5 | x | + 6 = 0 has no real roots.
It is a true declarative sentence, so it is a statement.

(ix) This sentence is a statement.
Without knowing the sentence, we cannot decide whether it is true or false. So, it is not a statement.

(x) Is the earth round?
It is an interrogative sentence, so it is not a statement.

(xi) Go!
It is an exclamatory sentence, so it is not a statement.

(xii) The real number x is less than 2.
We cannot decide whether this sentence is true or false without knowing the value of $x$. So, it is not a statement.

(xiii) There are 35 days in a month.
It is a false assertive sentence, so it is a false statement.

(xiv) Mathematics is difficult.
Mathematics could be easy for some people, so this sentence may or may not be true. So, it is not a statement.

(xv) All real numbers are complex numbers.
It is true because we can write a real number as $x+0i$. So, it is a true statement.

(xvi) The product of (−1) and 8 is 8.
It is an assertive sentence; therefore, it is a statement. But $-1\times 8=-8$; therefore, the statement is false.

Answer:

1) I won the trophy!
It is an exclamatory sentence, so it is not a statement.

2) Please fetch me a glass of water.
It is an imperative sentence. In other words, it can be expressed either as a request or as a command. Therefore, it not a statement.

3) Can you do this work for me?
It is an interrogative sentence, so it is not a statement.

Answer:

For given statement;
p: "The product of 3 and 4 is 9" 
~p: It is false that the product of 3 and 4 is 9. 

Hence, the correct answer is option A 

Answer:

For given statement; 
p: " A natural number is greater than zero".
~p: A natural number is not greater than zero.
∴ Not negation of p is the statement, It is false that a natural number is not greater than zero. 

Hence, the correct answer is option C. 

Answer:

For given statement; 
p: ''Earth revolves around the sun and moon is a satellite of earth''.
Since Earth revolves around the sun has no influence on the other statement. 
∴ The two statements are connected by and.

Hence, the correct answer is option D. 

Answer:

For the statement; 
p: ''If x > y, then x + a > y + a'' 
i.e. x > y ⇒ x + a > y + a
Converse of p is
x + a > y + a ⇒ x > y 
i.e. If x + a > y + a then x > y

Hence, the correct answer is option B. 

Answer:

For given statement; 
p: ''if sun is not shining, then sky is filled with clouds''.
Converse of p; if sky is filled with clouds, then sun is not shining. 

Hence, the correct answer is option A. 

Answer:

For given statement; 
p: if x² is not even, then x is not even. 
i.e. p: x² is not even ⇒ x is not even.
Converse of p; x is not even ⇒ x² is not even.
i.e. if x is not even, then x² is not even.

Hence, the correct answer is option B.

Answer:

None of the statement is a conjunction, because no two compound statement formed are joined. 

Hence, the correct answer is option D.

Answer:

p only if q is same p → q.

Hence, the correct answer is option C. 

Answer:

For given statement;
p: If Chandigarh is capital of Punjab, then 
q: Chandigarh is in India 
i.e. p ⇒ q 
Contrapositive form of p is; ~q ⇒ ~p. 
i.e. If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.
 
Hence, the correct answer is option A. 

Answer:

For given statement; 
p: If 7 is greater than 5, then 
q 8 is greater than 6
i.e. p ⇒ q is given. 
contrapositive form is ~q ⇒ ~p
i.e. If 8 is not greater than 6, then 7 is not greater than 5. 

Hence, the correct answer is option C. 

Answer:

For given statement; 
p: All triangle are equilateral triangles. 
~p: Not all triangle are equilateral triangles. or,
~p: It is false that all the triangle are equilateral triangles. or,
~p There exist a triangle which is not an equilateral triangle. 

Hence, the correct answer is option D. 

Answer:

p ⇒ (q ∨ ∼ q) is a tautological statement, if assertion is true in all possible interpretation. 

Hence, the correct answer is option A.

Answer:

Since (∼q) ∧ (p ∧ q) 
States that both q and ~q are occurring 
Simultaneously; which is a contradiction. 

Hence, the correct answer is option D. 

Answer:

For given statement; 
∼p ⇒ (p∧ ∼q) 
has contrapositive from given by 
∼(p∧ ∼q) ⇒ ∼ (∼p) 
i.e. (∼p∨ ∼(∼q)) ⇒ p   (By De-Morgan's law) 
i.e. (∼p ∨ q) ⇒ p

Hence, the correct answer is option D. 

Answer:

Given ~p = q
Then, (p ⇒ q) ∨ (q ⇒ p) 
is (p ⇒ ~p) ∨ (~p ⇒ p) 
i.e. both the possibility of occurrence is observed. 
Hence, statement is a tautology statement. 

Hence, the correct answer is option A.

Answer:

If truth value of p ∨ q is true 
i.e. either p is true or q is true or both p and q are true.
Then If p is true, ~p ∧ q is false.
If q is true; ~p ∧ q; cannot be judged.

Hence, the correct answer is option A. 

Answer:

p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r) is called

$\frac{\mathrm{distributive} \mathrm{law}}{once q \mathrm{and} r } \mathrm{since} \mathrm{it} \mathrm{distributes} P$

i.e. p ∧ (q ∨ r) ≅  (p ∧ q) ∨ (p ∧ r) is called distributive law.

Answer:

For 
~(p ∧ q) ≅ ~p ∨ ~q 
is called De Morgan's Law

Answer:

~(p ∨ ~q) = (~p ∧ ~(~q)) 
i.e. ~(p ∧ ~q) = (~p ∧ q)  (By De- Morgan's Law) 
i.e. ~p ∧ q is the answer 

Answer:

Given, (p ∨ ~q) ∧ q
          = (p ∧ q) ∨ (~q ∧ q) [By distributive law] 
          = (p ∧ q) ∨ ϕ
          = p ∧ q
Hence, (p ∨ ~q) ∧ q = p ∧ q.

Answer:

ans

Answer:

Given : ~(p ∧ q) ∨ r
Negation of above statement 
is ~(~(p ∧ q) ∨ r) 
i.e. ~(~(p ∧ q)) ∧ ~r (By Demorgan's Law) 
i.e. (p ∧ q) ∧ ~r
Hence, ~(p ∧ q) ∨ r is equivalent to (p ∧ q) ∧ ~r

Answer:

P ⇒ q is logically equivalent  to ~q ⇒ ~p 
i.e. either q is true. 
or q is false i.e. ~q ⇒ ~p 
∴ ~p is true 
∴ p ⇒ q is logically equivalent to q ∨ ~p
 i.e. ∼p ∨ q. 

Answer:

p ⇔ q given 
i.e. p is true if q is true 
or p is false if q is false
i.e. p ⇔ q is equivalent to (p ∧ q) ∨ (~p ∧ ~q) 
i.e. both p and q are true or false simultaneously
∴ p ⇔ q ≅ (p ∧ q) ∨ (~p ∧ ~q)

Answer:

Given, p ∨ ~q is false 
i.e. both p and ~q are false 
i.e. p is false but q is true 
i.e. q is true (T) 

Answer:

For given statement; 
p ⇒ ~q 
Contrapositive form is ~(~q)  ⇒ ~p
i.e. q ⇒ ~p 
∴ Contrapositive form p ⇒ ~q is q ⇒ ~p

Answer:

~[~p ∧ (p ⇔ q)) 
Since p ⇔ q is equivalent to if p than q and if q than p. 
or p ⇔ q is equivalent to  ~q ⇒ ~p and ~p ⇒ ~q 
∴ ~[~p ∧ {(~q ⇒ ~p) ∧ (~p ⇒ ~q)}]
= ~[~p ∧  (~p ⇒ ~q)] 
= ~[~p ∧ (~q)] (∴ ~p is True) 
= p ∨ q  (By De-Morgan's Law)

Answer:

For given statement,
p ⇒ p ∨ q
both the possibility are present 
∴ p ⇒ p ∨ q is a tautology 

Answer:

For given statement,
(p ⇔ q) ⇔ (~p ⇔ ~q)
both the possibility of statements is possible 
∴ (p ⇔ q) ⇔ (~p ⇔ ~q) is tautology

Answer:

For given statement;
p ∨ ~(p ⇒ ~q) 
i.e. p ∨ ~(p ⇒ ~q) 
i.e. p ∨ (p ⇒ ~q) 
i.e. p ∨ ~(p ⇒ ~q) is equivalent to p and q.

Answer:

(i) Negation of the given statement:
It is not true that Bangalore is the capital of Karnataka.
Or
Bangalore is not the capital of Karnataka.

(ii) Negation of the given statement:
It is not true that it rained on July 4, 2005.
Or
It did not rain on July 4, 2005.

(iii) Negation of the given statement:
It is not true that Ravish is honest.
Or
Ravish is not honest.

(iv) Negation of the given statement:
The earth is not round.
Or
It is not true that the earth is round.

(v) Negation of the given statement:
The sun is not cold.
Or
It is not true that the sun is cold.

Answer:

​(i) Negation of the given statement:
Some birds do not sing.
Or
There exists a bird that does not sing.

(ii) Negation of the given statement:
Some integers are not prime.
Or 
No even integer is prime.

(iii) Negation of the given statement:
All complex numbers are real numbers.

(iv) Negation of the given statement:
I will go to school.

(v) Negation of the given statement:
Both the diagonals of a rectangle do not have the same length.
Or
Both the diagonals of a rectangle have different lengths.

(vi) Negation of the given statement:
There exists a policeman who is not a thief.
Or 
At least one policeman is not a thief.

Answer:

​(i) The number x is not a rational number.
The number x is not an irrational number.
The statements in this pair are the negation of each other.

(ii) The number x is not a rational number.
The number x is an irrational number.​
The statements in this pair are not the negation of each other because both statements are the same. Both the statements convey that x is an irrational number.

Answer:

(i) ​p: For every positive real number x, the number (x − 1) is also positive.
 ~p: At least for one positive real number x, the number $(x-1)$ is not positive.

(ii) q: For every real number x, either x > 1 or x < 1.
 ~q: At least for one real number x, neither $x > 1$ nor $x < 1$.

(iii) r: There exists a number x such that 0 < x < 1.
   ~r: For every real number x, either $x \le 0$ or $x < 1$.

Answer:

​The given statements are not negation of each other because the negation of "$a+b=b+a$ is true for every real number a and b" is "There exist real numbers a and b for which $a+b\ne b+a$".