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Multiple choice validating statements proofs in mathematics mathematical reasoning maths

Determine whether the argument used to check the validity of the following statement is correct.
$p:$ If $x^{2}$ is irrational, then $x$ is rational'
The statement is true because the number $x^{2}=\pi^{2}$ is irrational, therefore $x=\pi$ irrational.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Here, the argument used is,


$x^2=\pi^2$ is irrational, therefore $x=\pi$ is irrational and, $p:$ " If 

$x^2$ is irrational, then $x$ is rational.

Let us take an irrational number given by $x=\sqrt n$,
where $n$ is a rational number.

Now, square both sides, we get,
$x^2=k$

Therefore, $x^2$ is a rational number, which contradicts our statement. 
Hence, the argument used to check validity of given statement is false.

Multiple choice validating statements proofs in mathematics mathematical reasoning maths

Tell if the following statement is true or false. In case give a valid reason for saying so
$p:$ If $x$ and $y$ are integers such that $x>y$. then $-x<-y$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given $x>y$


Multiply both sides by $-1$

$-x<-y$ $\therefore$ both statements true.

Multiple choice validating statements proofs in mathematics mathematical reasoning maths

If p and q are mathematical statements, then in order to show that the statement p and q is true, we need to show that:

  1. The statement p is true and the statement q is not true

  2. The statement p is false and the statement q is true.

  3. The statement p is true and the statement q is false

  4. The statement p is true and the statement q is true
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In logic, a conjunction (p and q) is true if and only if both individual component statements are true.

Multiple choice validating statements proofs in mathematics mathematical reasoning maths

The component statements are:

p: You are wet when it rains.

q: You are wet when you are in river.

The compound statement of these component statements using appropriate connective is:

  1. You are not wet when you are in river or it rains.

  2. You are wet when you are in river and it rains.

  3. You are wet when it rains and you are in a river

  4. You are wet when it rains or you are in a river.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The statement 'You are wet when it rains or you are in a river' uses the inclusive 'or' to connect the two conditions under which one becomes wet.

Multiple choice validating statements proofs in mathematics mathematical reasoning maths

Two pairs of statement are:
p: If a quadrilateral is a rectangle, then its opposite sides are equal.
q: If opposite sides of a quadrilateral are equal, then the quadrilateral is a rectangle.
The combined statement of these pairs using If and only if is:

  1. A quadrilateral is a rectangle if and only if its all sides are equal.

  2. A quadrilateral is a rectangle if and only if its opposite sides are equal.

  3. A quadrilateral is a square if and only if its opposite sides are equal.

  4. A quadrilateral is not a rectangle if and only if its opposite sides are equal.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

An 'if and only if' statement combines a conditional and its converse. Since the rectangle property is defined by opposite sides being equal, this is the correct biconditional.

Multiple choice validating statements proofs in mathematics mathematical reasoning maths

Name the technique used in the first step of the solution to the problem below :
Verify that 5 is irrational
Solution : Let us assume that 5 is rational

  1. Counter example

  2. Direct method

  3. By Contradiction

  4. Contrapositive method
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The proof technique that starts by assuming the negation of the statement to reach a contradiction is called proof by contradiction.