Tag: proofs in mathematics

Questions Related to proofs in mathematics

The dual of the statement $\left[ p\wedge \left( \sim q \right)  \right] \wedge \left( \sim p \right)] $ is

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

  1. $p\vee \left( \sim q \right) \vee \sim p$

  2. $\left( p\vee \sim q \right) \vee \sim p$

  3. $p\wedge \sim \left( q\vee \sim p \right) $

  4. none of these

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Correct Option: B

The contrapositive of the sentence $\sim p \rightarrow q$ is equivalent to

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

  1. $p \rightarrow \sim q$

  2. $q \rightarrow \sim p$

  3. $q \rightarrow p$

  4. $\sim p \rightarrow \sim q$

  5. $\sim q \rightarrow \sim p$

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Correct Option: E
Explanation:

For a conditional statement p → q, Its converse statement (q → p) and inverse statement (∼p → ∼q) are equivalent to each other. p → q and its contrapositive statement (∼q → ∼p) are equivalent to each other.

Write the inverse and contrapositive of the statement
"If two triangles are congruent, then their areas are equal."
$(a)$Inverse of the statement :
If two triangles are not congruent, then their areas are equal.
$(b)$Contrapositive of the statement:
If the areas of the two triangles are equal, then the triangles are congruent.

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

  1. $(a)False$ and $(b)$ False

  2. $(a)True$ and $(b)$ False

  3. $(a)False$ and $(b)$ True

  4. $(a)True$ and $(b)$ True

Show answer
Correct Option: A
Explanation:

"If two triangles are congruent, then their areas are equal."
$(a)$Inverse of the statement :
If two triangles are not congruent, then their areas are not equal.
$(b)$Contrapositive of the statement:
If the areas of the two triangles are not equal, then the triangles are not congruent.

What is the symbolic form and truth value of the following?
"If $4$ is an odd number, then $6$ is divisible by $3$." 
p: $4$ is an odd number.
q: $6$ is divisible by $3$.

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

  1. p$\rightarrow$q and $F$

  2. q$\rightarrow$p and $T$

  3. q$\rightarrow$p and $F$

  4. p$\rightarrow$q and $T$

Show answer
Correct Option: D
Explanation:

$p: 4$ is an odd number.
$q: 6$ is divisible by $3$.
Symbolic form: $p$ $\rightarrow$ $q$
$p$ is false and $q$ is true.
So, $F\rightarrow T$ is $T$.

Identify the Law of Logic
$\sim(\sim p) \equiv p$

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

  1. DeMorgan's Law

  2. Conditional Law

  3. Involution Law

  4. Complement Law

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Correct Option: C
Explanation:

Involution Law

This law states that if you negate a negation they effectively cancel each other out.
$\sim (\sim p)\equiv p$

Which of following is the negation of $(P \ \vee\sim Q).$

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

  1. $\sim P\vee Q$

  2. $\sim P\wedge Q$

  3. $\sim Q\wedge P$

  4. $\sim Q\vee P$

Show answer
Correct Option: B
Explanation:
 P  Q  $\sim P$  $\sim Q$  $P\vee \sim Q$ $\sim \left( P\vee \sim Q \right) $  $\sim P\wedge Q$ 
T   T  F  F  T  F  F
T F   F  T  T  F  F
F T   T  F  F  T  T
F F  T  T  F  T  F
Therefore, $\sim \left( P\vee \sim Q \right) $ is $\sim P\wedge Q$ 

Identify the Law of Logic
$p \wedge T \equiv T$
$p \vee F \equiv F$

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

  1. Complement Law

  2. Identity Law

  3. Involution Law

  4. Absorption Law

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Correct Option: B
Explanation:

Identity Law

Identity law observes how certain expression will behave when one of the terms is fixed
$p \vee F \equiv F$ and $p\wedge T\equiv T$

Is $(p\rightarrow q)\vee (q\rightarrow p)$  a tautology ?

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

  1. True

  2. False

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Correct Option: A
Explanation:
$p$ $q$ $(p\rightarrow q)$ $(q\rightarrow p)$ $(p\rightarrow q)\vee(q\rightarrow p)$
T T             T              T                               T
T F             F              T                               T
F T             T              F                               T
F F             T              T                               T               

The given statement is a tautology as the truth table has all the values as true in the output which is the property of tautology

Identify the Law of Logic
$(p \vee q) \vee r \equiv p \vee (q \vee r) \equiv p \vee q \vee r$

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

  1. Associative law

  2. Commutative Law

  3. Involution Law

  4. Conditional Law

Show answer
Correct Option: A
Explanation:

Associative Law

This law allows the removal of brackets from an expression and regrouping of the variables.
$(p\vee q)\vee r \equiv p \vee (q \vee r)\equiv p\vee q\vee r$

Identify the Law of Logic
$\sim(p \wedge q) \equiv \sim p \vee \sim q$

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

  1. Commutative Law

  2. DeMorgan's Law

  3. Complement Law

  4. Conditional Law

Show answer
Correct Option: B
Explanation:
Given 
$\sim (p\wedge q)=\sim p \vee \sim q$

It is Demorgan's law 
according to the if we take transpose or negation of any quatity then all the relation get opposite