Tag: maths

Questions Related to maths

Let p and q be any two logical statements and $r : p \rightarrow (\sim p \vee q)$. If r has a truth value F, then the truth values of p and q are respectively

maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

  1. F, F

  2. T, T

  3. F, T

  4. T, F

Show answer
Correct Option: D
Explanation:
p q $\sim$p $\sim$ p $\vee$ q r
T T F T T
F F T T T
T F F F F
F T T T T

$\therefore$ Clearly from above able, If r has a truth value F, then the truth values of p and core T and F respectively.

State, whether the is given the statement, is True or False.
$\sim [(p \vee \sim q) \rightarrow (p \wedge \sim q)] \equiv (p \vee \sim q) \wedge (\sim \vee q)$

maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
 $p$  $q$  $\sim q$  $(p\vee \sim q)$  $(p\wedge \sim q)$ $(p\vee \sim q)\rightarrow (p\wedge \sim q)$  $\sim[(p\vee \sim q)\rightarrow (p\wedge \sim q)]$
 $T$  $T$  $F$  $T$  $F$  $F$  $T$
 $T$  $F$  $T$  $T$  $T$  $T$   $F$
 $F$  $T$  $F$  $F$  $F$  $T$   $F$
 $F$  $F$  $T$  $T$  $F$  $F$ $T$
 $p$  $q$  $\sim q$  $(p\vee \sim q)$ $\sim p$  $(\sim p \vee q)$ $(p\vee \sim q)\wedge (\sim p \vee q)$
 $T$  $T$  $F$  $T$  $F$  $T$  $T$
 $T$  $F$  $T$  $T$  $F$  $F$  $F$
 $F$  $T$  $F$   $F$  $T$  $T$  $F$
 $F$  $F$  $T$  $T$  $T$  $T$  $T$

$\sim (p \vee q) \vee (\sim p \wedge q) \equiv ?$

maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

  1. $\sim q$

  2. $q$

  3. $\sim p$

  4. $p$

Show answer
Correct Option: C
Explanation:

$\sim (p\vee q)\vee (\sim p \wedge q)$
$=(\sim p \wedge \sim q) \vee (\sim p \wedge q)$ De'Morgan Law
$=(\sim p)\wedge (\sim q\vee q)$ Distributive Law
$=(\sim p)\wedge T$ Negation Law
$=\sim p$ Identity law

State whether the following statements is True or False?
$p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)$

maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
given statement 
$p\leftrightarrow q=(p\wedge q)\vee (\sim p\wedge \sim q)$
taking RHS
$(p\wedge q)\vee (\sim p\wedge \sim q)$
$(p\wedge q)\vee (\sim(p\wedge  q))$
$(p\wedge q)\wedge(p\wedge  q)$
$p\leftrightarrow q$
It is true

Let p,q be statements. Negation of statement $p \leftrightarrow  ~ q$, is

maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

  1. $~ q \rightarrow p$

  2. $ ~ p v q$

  3. $p \leftrightarrow q$

  4. $p \rightarrow q$

Show answer
Correct Option: A
Explanation:
p q ~q $p \leftrightarrow ~q$ $~(p \leftrightarrow ~q)$ $~q \rightarrow p$ $~p v q$ $p \leftrightarrow q$ $p \rightarrow q$
T T F F T T T T T
T F T T F T F F F
F T T T F T T F T
F F F F T F T T T

Which point on the number line most likely represent $-2\cfrac { 5 }{ 8 } $?

maths parts and whole finding the whole when a fraction is given multiplication of a fraction multiplication of a fractions

  1. On the left of $-3$

  2. On the right of $-2$

  3. Between $-2$ and $-3$

  4. In the middle of $-2$ and $-3$

Show answer
Correct Option: C
Explanation:
$-2\cfrac { 5 }{ 8 } $

$-2\cfrac { 5 }{ 8 } =(-1)2\cfrac { 5 }{ 8 } $

$=-1\times \cfrac { 21 }{ 8 } $

$=-2.625$

It lies in between $-3$ and $-2$

$A,B,C$ and $D$ are all different digits between $0$ and $9$. If $AB+DC=7B\ (AB,DC$ and $7B$ are two digit numbers), then the value of $C$ is

maths number systems existence of irrational numbers irrational numbers properties of irrational numbers

  1. $0$

  2. $1$

  3. $2$

  4. $3$

  5. $5$

Show answer
Correct Option: A

If $\sqrt{a}$ is an irrational number, what is a? 

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Rational

  2. Irrational

  3. $0$

  4. Real

Show answer
Correct Option: A
Explanation:

Consider the given irrational number$\sqrt{a}$ ,

Definition  of rational number- which number can be write in the form of $\dfrac{p}{q}$ but $q\ne 0$ is called rational number.

Hence, $a=\dfrac{a}{1}$

That why  $a$ is rational number

 

Hence, this is the answer.

Which of the following is irrational

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $\sqrt {\dfrac{4}{9}} $

  2. $\dfrac{4}{5}$

  3. $\sqrt 7 $

  4. $\sqrt {81} $

Show answer
Correct Option: C
Explanation:
A $=\sqrt{\dfrac{4}{9}}=\dfrac{2}{3}$         Rational

B $=\dfrac{4}{5}$                       Rational

C $=\sqrt7$                     Irrational

D $=\sqrt{81}=9$          Rational

The number $23+\sqrt{7}$ is

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Natural number

  2. Irrational number

  3. Rational number

  4. None of these

Show answer
Correct Option: B
Explanation:

As we've $\sqrt{7}$ is an irrational number and $23$ is a rational number then the sum of an irrational number and a rational number is again an irrational number.