Cartesian Product

Q.

How do you evaluate ${}^{10}C_3$?

Q.

Let $A=\left\{x,y,z\right\}$ and $B=\left\{a,b,c,d\right\}$. which one of the following is not a relation from $A$to $B$

1. $\left\{\left(x,a\right),\left(x,c\right)\right\}$

2. $\left\{\left(y,c\right),\left(y,d\right)\right\}$

3. $\left\{\left(z,a\right),\left(z,d\right)\right\}$

4. $\left\{\left(z,b\right),\left(y,b\right),\left(a,d\right)\right\}$

5. $\left\{\left(x,c\right)\right\}$

Q.

If $S_1=\left\{1,2,3,\dots ..20\right\},S_2=\left\{a,b,c,d\right\},\text{and}{S}_3=\left\{b,d,e,f\right\}$. The number of elements of $\left(S_1\times S_2\right)\cup \left(S_1\times S_3\right)$ is

1. $100$

2. $120$

3. $140$

4. $40$

Q.

If A = {1, 2, 4}, B = {2, 4, 5} C = {2, 5}, then $(A-B)\times (B-C)$ is

1. {(1, 2), (1, 5), (2, 5)}

2. {(1, 4)}

3. None of these

4. (1, 4)

Q. Let $Y=\{1,2,3,4,5\},A=\{1,2\},B=\{3,4,5\}.$ If $A\times B$ denotes the cartesian product of sets $A$ and $B,$ then $(Y\times A)\cap (Y\times B)$ is

1. $Y$
2. $B$
3. $A$
4. Null set

Q.

A relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows : $\{x,y\}ϵR\iff x$ is relatively prime to y. Express R as a set of ordered pairs and determine its domain and range.

Q.

Let f, g: R → R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and.

Q.

If A = {1, 2, 3}, B = {4} and C = {5}, then verify that :

(i) $A\times (B\cup C)=(A\times B)\cup (A\times C)$

(ii) $A\times (B\cap C)=(A\times B)\cap (A\times C)$

(iii) $A\times (B-C)=(A\times B)-(A\times C)$

Q.

Let A and B be two sets. Show that the sets $A\times B$ and $B\times A$ have an element in common if the sets A and B have an element in common.

Q.

If A = {-1, 1}, find $A\times A\times A$.

Q. If $a,b,c$ be three unequal positive quantities in H.P., then

1. $a^{100}+c^{100}>2b^{100}$
2. $a^3+c^3>2b^3$
3. $a^5+c^5>2b^5$
4. $a^2+c^2>2b^2$

Q. Let $A,B,C$ are three sets such that $n(A\cap B)=n(B\cap C)=n(C\cap A)=n(A\cap B\cap C)=2$, then $n\left(\right(A\times B)\cap (B\times C\left)\right)$ is equal to

Q. A lot contains $50$ defective and $50$ non-defective bulbs. Two bulbs are drawn at random one at a time with replacement. The events $A,B,C$ are defined as :
$A:$ $1$st bulb is defective
$B:$ $2$nd bulb is non-defective
$C:$ both are defective or both are non-defective, then

1. $A,B,C$ are pairwise as well as mutually independent.
2. $A,B,C$ are pairwise independent but mutually not independent.
3. $A,B,C$ are mutually independent but pairwise not independent.
4. None of these.

Q.

If two sets $A$ and $B$ are having $43$elements in common, then the number of elements common to each of the sets$A\times B$and $B\times A$is

1. ${43}^2$

2. $2^{43}$

3. ${43}^{43}$

4. $2^{86}$

Q.

The Cartesian product of two sets $P$ and $Q$, i.e., $P\times Q=\varnothing$, if

1. either $P$ or $Q$ is the null set.

2. neither $P$ nor $Q$ is the null set

3. Both (A) and (B)

4. None of these

Q.

If A = {1, 2}, form the set $A\times A\times A$

Q. Let $X=\{x:x\in N\text{and}-3<x<4\}$ and $Y=\{x:x\in N\text{and}{x}^2+x-6\le 0\}.$ Then the number of elements in $(X\times Y)\cap (Y\times X)$ is

1. $0$
2. $1$
3. $2$
4. $4$

Q.

Find the sets $A$ and $B$ if $\frac{A}{B}=\left\{1,5,7,8\right\},\frac{B}{A}=\left\{2,10\right\}$ and $A\cap B=\left\{3,6,9\right\}$. Are they unique?

Q. Consider functions f and g such that composite gof is defined and is one-one. Are f and g both necessarily one-one.

Q.

If A = {2, 3}, B = {4, 5} and C = {5, 6}, find $A\times (B\cap C),(A\times B)\cup (A\times C).$

Q.

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(i) $A\times (B\cap C)$

(ii) $(A\times B)\cap (A\times C)$

(iii) $A\times (B\cup C)$

(iv) $(A\times B)\cup (A\times C)$

Q. If $A$ and $B$ are two sets such that $A\times B=\left\{\right(1,1),(1,2),(1,3),(2,1),(2,2),(2,3\left)\right\}$, then $B$ is

1. $\{1,2,3\}$
2. $\{1,2\}$
3. $\{2,3\}$
4. none of these

Q. The figure shows a relationship between the sets $P$ and $Q$. Write this relation.

$\left(i\right)$ In set builder form
$\left(ii\right)$ In roster form
What is its domain and range ?

Q.

Product of two odd functions is

1. Even function

2. Odd function

3. Neither even nor odd

4. Cannot be determined

Q. Let a relation R be defined by R = {(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)}. The relation $R^{-1}$ o R is given by

1. {(1, 1), (4, 4), (7, 4), (4, 7), (7, 7)}
2. {(1, 1), (4, 4), (4, 7) (7, 4), (7, 7) (3, 3)}
3. {(1, 5), (1, 6), (3, 6)}
4. None of the above

Q. Let $A=$ {$1,2,3,4,...14$}. Define a relation $R$ from $A$ to $A$ by $R=$ {$(x,y):3x-y=0,\text{where}x,yϵA$}. Write down its domain, co-domain, and range.

Q.

If A and B are two sets having 3 elements in common. If n(A) = 5 and n(B) = 4, find $n\left[\right(A\times B)$ and $n(A\times B)\cap (B\times A)]$.

Q. R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x – 3. The relation $R^{-1}$ is

1. {(11, 8), (13, 10)}
2. {(8, 11), (10, 13)}
3. {(8, 11), (9, 12), (10, 13)}
4. None of the above

Q. Let $A$ be a non-empty set such that $A\times A$ has $16$ elements among which three elements are found to be $(a,b),(b,c)$ and $(c,d)$ and $I_A$ be the identity relation on $A,$ then

1. $A=\{a,b,c\}$
2. $I_A=\left\{\right(a,a),(b,b),(c,c\left)\right\}$
3. $A=\{a,b,c,d\}$
4. $I_A=\left\{\right(a,a),(b,b),(c,c),(d,d\left)\right\}$

Q. If $A=\{3,4,5\},B=\{1,2,3\}$ then

1. $n(A\times B)=n(B\times A)$
2. Total number of relations from $A$ to $B$ is $2^8$
3. $n\left(\right(A\times B)\times (B\times A\left)\right)=81$
4. $(A\times B)\cap (B\times A)=\left\{\right(3,3\left)\right\}$