Tag: principle of mathematical induction

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Questions Related to principle of mathematical induction

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

The compound proposition which is always false is:

  1. $\left(p \rightarrow q\right)\leftrightarrow \left( \sim q \rightarrow \sim p \right) $
  2. $\left[ \left( p\rightarrow q \right) \wedge \left( q\rightarrow r \right) \right]\rightarrow \left( p\rightarrow r \right) $
  3. $\left( \sim p\vee q \right) \leftrightarrow \left( p\wedge \sim q \right) $
  4. $p \rightarrow \sim p$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A compound proposition is always false if it is a contradiction. Option C equates (not p or q) with (p and not q), which are exact opposites, meaning their biconditional is always false.

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

If  $p$ and  $q$ are two simple proposition then  $p \rightarrow q$  is false when

  1. $p \text { is true and } q \text{ is true}$
  2. $p \text { is false and } q \text{ is true}$
  3. $p \text { is true and } q \text{ is false}$
  4. both $p$ and $q$ are false
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A conditional statement p implies q is false only in the single case where the hypothesis p is true and the conclusion q is false. In all other cases, the implication evaluates to true.

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

Let  $p :$  Mathematics is interesting and let  $q:$  Mathematics is difficult, then the symbol  $p\wedge q$  means

  1. Mathematics is interesting implies that Mathematics is difficult

  2. Mathematics is interesting implies and is implied by Mathematics is difficult

  3. Mathematics is interesting and Mathematics is difficult

  4. Mathematics is interesting or Mathematics is difficult
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$'\Lambda '$ stands for logical and 

$\therefore$    $p\Lambda q$ means 
Mathematics is interesting and Mathematics is difficult.

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

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

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The original statement has mismatched brackets: [p ∧ (¬q)] ∧ (¬p)] - this is grammatically incorrect. Assuming the intended statement is p ∧ (¬q) ∧ (¬p), its dual is formed by replacing ∧ with ∨: p ∨ ¬(q) ∨ ¬(p). Using De Morgan's law, ¬(q) ∨ ¬(p) ≡ ¬(q ∧ p), so the dual can also be written as (p ∨ ¬q) ∨ ¬p, which matches Option B.

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

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

  1. $p \rightarrow \sim q$
  2. $q \rightarrow \sim p$
  3. $q \rightarrow p$
  4. $\sim p \rightarrow \sim q$
  5. $\sim q \rightarrow \sim p$
Reveal answer Fill a bubble to check yourself
E Correct answer
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.

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

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.

  1. $(a)False$ and $(b)$ False
  2. $(a)True$ and $(b)$ False
  3. $(a)False$ and $(b)$ True
  4. $(a)True$ and $(b)$ True
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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

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 $3$.

  1. p$\rightarrow$q and $F$
  2. q$\rightarrow$p and $T$
  3. q$\rightarrow$p and $F$
  4. p$\rightarrow$q and $T$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $T$.

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

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

  1. $\sim P\vee Q$
  2. $\sim P\wedge Q$
  3. $\sim Q\wedge P$
  4. $\sim Q\vee P$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $\sim P\wedge Q$