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

Which of the following is always true ? 

  1. $\left( {p \to q} \right) \cong \left( { \sim q \to \sim p} \right)$
  2. $ \sim \left( {p \vee q} \right) \cong \left( { \sim p \vee \sim q} \right)$
  3. $ \sim \left( {p \to q} \right) \cong \left( {p \vee \sim q} \right)$
  4. $ \sim \left( {p \wedge q} \right) \cong \left( { \sim p \wedge \sim q} \right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know that 


$p\rightarrow q\equiv \sim p\wedge q$

$\sim (p\rightarrow q)\equiv \sim (\sim p\wedge q)$

$\sim (p\rightarrow q)\equiv p\vee \sim q$               (De morgan's law)

$C$ is correct

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

$p \leftrightarrow q \equiv  \sim \left( {p\Delta  \sim q} \right)\Delta  \sim \left( {q\Delta  \sim p} \right)$

  1. True

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

The expression uses the delta symbol, which usually denotes XOR. The identity p <-> q is equivalent to the negation of XOR, and the expression provided simplifies to this identity.

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

Let $p$ and $q$ be two statements, then $ \sim ( \sim p \wedge q) \wedge (p \vee q)$ is logically equivalent to 

  1. $q$
  2. $p\vee q$
  3. $p$
  4. $p\vee \sim q$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using De Morgan's laws, not(not p and q) becomes p or not q. Distributing this with (p or q) yields p or (not q and q), which simplifies to p or False, resulting in p.

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

$ \sim (p \wedge q) \to ( \sim p \vee ( \sim p \vee q))$  is equivalent to 

  1. $p \vee \sim q$
  2. $p \wedge \sim q$
  3. $ \sim p \vee q$
  4. $ \sim p \wedge q$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An implication A to B is equivalent to not A or B. Here, A is not(p and q), so not A is (p and q). B is not p or q. Thus, the expression becomes (p and q) or not p or q, which simplifies using absorption laws to not p or q.

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.