Compound Statement

Q. The Boolean expression $(p\wedge \sim q)\vee q\vee (\sim p\wedge q)$ is equivalent to

1. $\sim p\wedge q$
2. $p\wedge q$
3. $p\vee q$
4. $p\vee \sim q$

Q. The logically equivalent proposition of $p\iff q$ is

1. $(p\wedge q)\Rightarrow (q\vee p)$
2. $(p\wedge q)\vee (p\wedge q)$
3. $(p\Rightarrow q)\vee (q\Rightarrow p)$
4. $(p\wedge q)\vee (q\Rightarrow p)$

Q. The false statement among the following is

1. $p\wedge (\sim p)$ is a contradiction
2. $(p\to q)\iff (\sim q\to \sim p)$ is a contradiction
3. $\sim (\sim p)\iff p$ is a tautology
4. $p\vee (\sim p)$ is a tautology

Q. Is the following collection, a set? Justify your answer:

(x) The collection of prime integers

Q. For the statement "A mixture of alcohol and water can be separated by chemical methods" which among the following option is correct

1. It is compound statement having connective ${}^{\prime \prime }O{R}^{\prime \prime }$
2. It is compound statement having connective ${}^{\prime \prime }AN{D}^{\prime \prime }$
3. It is a simple statement
4. It is not a statement

Q. The logical statement $\left[\right(p\wedge q)\to p]\to (q\wedge \sim q)$ is

1. a tautology
2. a contradiction
3. neither a tautology nor a contradiction
4. equivalent to $p\vee q$

Q. The negation of the statement "Some even integers are prime" is/are:

1. Some even integers are not prime
2. No even integer is prime
3. It is true that some even integers are prime
4. It is false that some even integers are prime

Q. Which among the following statements have truth value as FALSE
$A)13$ is odd number and prime number
$B)23$ is even number and prime number
$C)$ Volume of sphere is $\frac{4}{3}\pi r^3$ cubic units and surface area is $\pi r^2$ sq. units

1. $B$ and $C$
2. $A$ and $B$
3. $A$ and $C$
4. $A,B$ and $C$

Q.

A simple statement can be a combination of 2 or more simple statements

1. False

2. True

Q.

A shopkeeper earns a profit of $1Rs$ by selling one pen and incurs a loss of $40paise$ per pencil while selling pencils of her old stock.

In the month she earns neither profit nor loss. If she sold $70pens$, how many pencils did she sell?

Q.

The numerator of a fraction is $3$‘ less than its denominator if ’$2$‘ is added to both the numerator and denominator, then the sum of the new fraction and the original fraction is ’$\frac{29}{20}$. Find the original fraction.

Q. Let $p,q,r$ denote the arbitary statements then the logical equivalance of the statement $p\Rightarrow (q\vee r)$ is

1. $(p\vee q)\Rightarrow r$
2. $(p\Rightarrow q)\vee (p\Rightarrow r)$
3. $(p\Rightarrow \sim q)\wedge (p\Rightarrow r)$
4. $(p\Rightarrow q)\wedge (p\Rightarrow \sim r)$

Q. The statement $\sim (p\leftrightarrow \sim q)$ is

1. equivalent to $p\leftrightarrow q$
2. a tautology
3. a fallacy
4. equivalent to $\sim p\leftrightarrow q$

Q. The negation of the statement: ''It is raining and it is cool'' is

1. It is raining and it is not cool.
2. It is not raining or it is not cool.
3. It is not raining and it is cool.
4. Neither it is raining nor it is cool.

Q. If the Boolen expression $(p\Rightarrow q)\iff (q\ast (\sim p\left)\right)$ is a tautology, then the Boolean expression $p\ast (\sim q)$ is equivalent to:

1. $p\Rightarrow \sim q$
2. $p\Rightarrow q$
3. $q\Rightarrow p$
4. $\sim q\Rightarrow p$

Q.

Determine whether the following compound statements are true or false :

(i) Delhi is in India and 2 + 2 = 4

(ii) Delhi is in England and 2 + 2 = 4.

(iii) Delhi is in India and 2 + 2 = 5.

(iv) Delhi is in England and 2 + 2 = 5.

Q. Convert compound sentence to simple sentence:
He is rich, yet he is unhappy.

1. He is rich but also unhappy.
2. He is rich and unhappy.
3. Even though he is rich he is unhappy.
4. Despite being rich he is unhappy.

Q. The Boolean expression $\left(\right(p\wedge q)\vee (p\vee \sim q\left)\right)\wedge (\sim p\wedge \sim q)$ is equivalent to :

1. $p\wedge q$
2. $(\sim p)\wedge (\sim q)$
3. $p\vee (\sim q)$
4. $p\wedge (\sim q)$

Q. The Boolean expression $(p\wedge \sim q)\vee q\vee (\sim p\wedge q)$ is equivalent to

1. $\sim p\wedge q$
2. $p\wedge q$
3. $p\vee q$
4. $p\vee \sim q$

Q. "11 is the fifth prime number."This is a statement.

1. simple
2. compound
3. none

Q. If $p:x$ is odd ; $q:x^2$ is odd, then "$x$ is odd and $x^2$ is not odd" is represented as

1. $p\vee q$
2. $p\wedge q$
3. $(\sim p)\vee q$
4. $p\wedge (\sim q)$

Q.

Write the component statements of the following compound statements and check whether the compound statement is true or false :

(i) To enter into a public library children need an identity card from the school or a letter from the school authorities.

(ii) All rational numbers are real and all real numbers are not complex.

(iii) Square of an integer is positive or negative.

(iv) x = 2 and x = 3 are the roots of the equation $3x^2-x-10=0$.

(v) The sand heats up quickly in the sun and does not cool down fast at night.

Q. If $x_1,x_2,x_3$ as well as $y_1,y_2,y_3$ are in G.P. with same common ratio, then the points $P\left(x_1,y_1\right)$, $Q\left(x_2,y_2\right)$ and $R\left(x_3,y_3\right)$

1. lie on a straight line
2. lie on an ellipse
3. lie on a circle
4. are vertices of a triangle

Q. Let A = {3, 5, 7}, B = {2, 6, 10} and R be a relation from A to B defined by R = {(x, y) : x and y are relatively prime}. Then, write R and R⁻¹.

Q. From the context of the two sentences given below form a single sentence.
The tired old woman was unable to go any further. She returned home.

Q. Convert compound sentence to simple sentence:
She is not only pretty but also clever.

1. No change
2. She is pretty but also clever.
3. Besides being pretty, she is clever.
4. She is pretty and clever.

Q.

If p: Arjun is the fastest q: Azad is the captain. Then which of the following denotes the compound statement: "Arjun is the fastest OR Azad is not the captain”

1. 

2. 

3. 

4. 

Q. Write the compound statement of the following compound statement and check whether the compound statement is true or false : "Zero is less than every positive integer and every negative integer"

Q. Convert the compound sentence to a simple sentence:
He was guilty and went into the hiding.

1. He went into the hiding because he was guilty.
2. Being guilty, he went into the hiding.
3. Since he was guilty he went into the hiding.
4. No change

Q. Given P : 25 is a multiple of 5, q : 25 is a multiple of 8. Write the compound statement connecting these two statements with "and", "or" in ${60}^{th}$ cases. Check the validity of the statement.