Tag: proofs in mathematics

Quizzes (2) Questions (121)

Questions Related to proofs in mathematics

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 $\sim P\wedge Q$ 
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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
Multiple choice maths proofs in mathematics implications principle of mathematical induction different forms of theoretical statements

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

  1. Associative law

  2. Commutative Law

  3. Involution Law

  4. Conditional Law
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Associative Law

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

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

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

  1. Commutative Law

  2. DeMorgan's Law

  3. Complement Law

  4. Conditional Law
Reveal answer Fill a bubble to check yourself
B Correct answer
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