## Skill Check III: IITJEE Foundation Maths

State whether the following statements are true or false:

1. If $A = \{ x | x = 5n, 5 < n < 10, n \in N\}$, then $n(A)=4$
2. If $n(A) = n(B)$, then $set A \leftrightarrow set B$
3. If Set $A= \{ x | x \in N, x<3\}$, then A is a singleton set.
4. The intelligent students of class VIII form a set.
5. The students passing the half-yearly exams in Class VIII B of school is a set.
6. $A = \{ x | x = p^{3}, p<4, p \in \mathcal{N}\}$ and $\{ x|x = m^{2}, m < 4, m \in \mathcal{N} \}$ are overlapping sets.
7. $A = \{ x | x \in \mathcal{N}\}$ is a subset of $B = \{ x | x in \mathcal{Z}\}$
8. If we denote the universal set as $\Omega = \{ p,q,r,s,t,u,v\}$ and $A = \{ q,u,s,t\}$, then $\overline{A} = \{ p, r, v\}$
9. $A = \{ x | x =2p, p \in \mathcal{N}\}$ and $B = \{ x | x =3p, p \in \mathcal{N}\}$ are disjoint sets.
10. If $A = \{ 2,3,4\}$, then $P(A)= \{ \phi, A, \{ 2\}, \{ 3\}, \{ 4 \}, \{ 2,3\}, \{ 2,4\}, \{ 3,4\}\}$ where $P(A)$ is the power set of set A.

II. If C is a letter in the word down all the subsets of C.

III. Write down the complements of all the 8 subsets of set C above.

IV. If $Q = \{ x : x =a^{2}+1, 2 \leq a \leq 5\}$, what is the power set of Q?

V. If $x = \{ x | x<20, x \in \mathcal{N}\}$, and if $A = \{x | x = 2a, 3 < a < 8, a \in \mathcal{N} \}$, and if $B = \{ x | x = 3b, b < 5, b \in \mathcal{N}\}$, and if $C = \{x | x = c+1, 5 < c < 15, c \in \mathcal{N} \}$, then find : (i) $n(B)$ (ii) $n(C)$ (iii) $\overline{A}$ (iv) $\overline{B}$ (v) $P(B)$

VI. If $A = \{ x| x \in \mathcal{N}, 3 < x < 10\}$, and if $B = \{ x| x =4a-1, a<5, a \in \mathcal{N}\}$ and if $C = \{ x | x = 3a+2, a<7, a \in \mathcal{N}\}$, then confirm the following: (i) the commutative property of the unions of sets B and C (ii) the commutative property of intersection of two sets A and C (iii) the associative property of the union of the sets A, B and C (iv) the associative property of intersection of sets A, B and C.

VII. If $A = \{ x | x \in \mathcal{N}, 4 \leq x \leq 12\}$, and $B = \{ x| x = a+1, a<8, a \in \mathcal{N}\}$, and $C= \{ x| x =2n, 1 < n <7, n \in |mathcal{N}\}$, then find (i) $A-B$ (ii) $B-C$ (iii) $B \bigcap C$ (iv) $A - (B \bigcap C)$ (v) $B - (A \bigcap C)$ (vi) $A-C$ (vii) $A- (B-C)$ (viii) $A- (B \bigcup C)$

VIII. If $\xi = \{ x | x \hspace{0.1in}is \hspace{0.1in} a \hspace{0.1in} letter \hspace{0.1in}of \hspace{0.1in}the \hspace{0.1in} English \hspace{0.1in} alphabet \hspace{0.1in} between \hspace{0.1in} but \hspace{0.1in} not \hspace{0.1in} including \hspace{0.1in} d \hspace{0.1in} and \hspace{0.1in} o\}$, and let $A = \{ l, m , n\}$ and let $B= \{ e,f,g,h,i,j,k,l\}$, and let $C = \{ j,k,l,m\}$, find (i) $\overline{A} \bigcup \overline{B}$ (ii) $\overline{B} \bigcap \overline{C}$ (iii) $A \bigcap C$ (iv) $B - (A \bigcap C)$ (v) $\overline{B-A}$ (vi) Is $(B-C) \subset (B-A)$? (vii) Is $\overline{A} \bigcap \overline{B} = \phi$?

IX. All 26 customers in a restaurant had either drinks, snacks, or dinner. 18 had snacks, out of which 6 had only snacks, 4 had snacks and drinks but not dinner, 2 had drinks and dinner but not snacks, and 3 had snacks and dinner but not drinks. If 14 customers had drinks, find (i) how many customers had all three — drinks, snacks as well as dinner. (ii) how many customers had dinner but neither snacks nor drinks (iii) how many customers had only drinks.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06