Sets

58. Let A = {a}, B = {a, b}, C = {a, c}

So, A$\cap$B = {a} $\cap$ {a, b} = {a}

A$\cap$C = {a} $\cap$ {a, c} = {a}

i.e., A$\cap$B = A$\cap$C = {a}

But B ≠C. as b$\cancel{\in }$B but $b\cancel{\ne }C$ vice-versa

57. (i) We know that,

A $\subset$A

(A $\cap$B)$\subset$ A

[A$\cup$ (A $\cap$B)] $\subset$ (A)

[A $\cup$ (A $\cap$B)] $\subset$A

and also

A$\subset$ [A $\cup$ (A $\cap$B)]

So, A$\cup$ (A$\cap$ B) = A.

(ii) A$\cap$ (A$\cup$ B) = (A$\cap$ A) $\cup$ (A$\cap$ B) [By distributive law]

= A$\cup$ (A$\cap$ B)

= A as (A $\cap$B) $\subset$A

56. Here,

(A$\cap$B)$\cup$ (A - B) = (A$\cap$B)$\cup$ (A$\cap$ B') as (A -  B) = A$\cap$ B'

= A $\cap$ (B$\cup$ B') [ by converse of distributive law]

= A $\cap$U [ B $\cup$B' = U, sample space set or universal set]

= A

And (A $\cup$ (B - A) = A $\cup$ (B$\cap$ A') [as B -  A = B$\cap$ A']

= (A$\cup$B) $\cap$ (A$\cup$A')

= (A$\cup$B)$\cap$ U [ A$\cup$A' = U, universal set]

= A$\cup$ B.

55. Let A = {a}, B = {b}.

So, P (A) = $\{?,\{a\left\}\right\}B(A)=\{?,(b\left\}\right\}$

So, P (A) $\cup$P {B} = $\left\{?,\left\{a\right\},\left\{b\right\}\right\}$ ______ (1)

Now, A$\cup$B = {a, b}.

P (A$\cup$B) = $\left\{?,\left\{a\right\},\left\{b\right\},\{a,b\}\right\}$ ____ (2)

So. From (1) and (2) we see that,

P (A)$\cup$ P (B) ≠P (A$\cup$B)

54. Given, P (A) = P (B) where P is power set

Let x$\cancel{\in }$A.

Then, {x} $\subset$P (A)$\subset$ P (B)

i.e., x$\cancel{\in }$ B

A$\subset$ B

Similarly, B $\subset$A

A = B

53. Given, A $\subset$B.

Let x$\cancel{\in }$ C - B then x $\cancel{\in }$C but X$B$ .

However, A $\subset$B, elements of B should have elements of A

i.e., X$B$ 

So, x $\cancel{\in }$C A i.e., x$\cancel{\in }$ C but X$A$

C - B $\subset$C-  A

52. (i) Let A-  B≠$?$ , our assumption.

i.e., x$\in$ A But x≠ B where x is an element.

But as A B, the above condition of assumption is wrong if A $\subset$B then A -  B =$?$

(ii) Let x$\in$ A.

As A - B =$?$ we can say that x$\in$ B because if $x\cancel{\in }B,$ A - B ≠$?$

if A - B =$?$ then A $\subset$B.

(iii) We know that,

B $\subset$A$\cup$ B always true

Let x$\in$A$\cup$B i.e., x$\in$ A or x$\in$ B.

As A $\subset$B,

If x $\in$A then x$\in$ B, all elements of A are among the elements of B

So, (A$\cup$B) = B

(iv) We know that,

(A$\cap$ B) $\subset$A as A$\subset$ B.

Let x $\in$A then x $\in$B

So, x $\in$(A$\cap$ B)

i.e., A$\subset$ (A$\cap$ B)

So, A = (A $\cap$B)

Hence, A$\subset$ B

A - B = $?$.

A$\cup$ B = B

A $\cap$B = A.

i.e., the 4 conditions are equivalent.

51. Let x$\in$b

As B$\subset$ A$\cup$ B we can write

Let x $\in$A $\cup$B.

as A $\cup$B = A $\cup$C.

x$\in$ A$\cup$ C.

i.e., x$\in$ A or x $\in$C

when x$\in$ A, and x$\in$ B,

x$\in$ A$\cap$ B

But A$\cap$ B = A $\cap$C

So, x$\in$ A $\cap$C

i.e., x$\in$ A and x$\in$ C

x $\in$C

when, x$\in$ C

as x $\in$B and x $\in$C

So, B $\subset$C

Similarly, C $\subset$B

So, B = C

50. (i) False Let A = {a}, a $\in$A then B = {a}, b} I e,  $a\cancel{\in B.}$

(ii) False. Let A. = {a}, if A $\subset$B, B = {a, b} and B $\in$C I e, C = {a, b}, c} I e, A = {a}. 

(iii) True. Let x$\in$A, if A$\subset$ B then x$\in$B and if B$\subset$ C, x$\in$ C I e, elements of A are also elements of C. A$\subset$ C.

(iv) False. Let A = {a} and B = {b} then A$\not\subset$$B$ . Let C = {a, c} then B$\not\subset$$\subset$ but a $\in$A and a c, i.e., A$\subset$ C.

(v) False. Let A = {a} and B = {b} so, A$\not\subset$$B$ i.e., A$B$ .

(vi) True. Let A$\subset$ B such that y$\in$ B i.e., y A But x$\in$$B$ and suppose x$\in$A.

Then by above definition,  A $\subset$B i.e., x$\in$ B and x$\in$ A which is not the case

49. A = {x: xR and x satisfy x² - 8x + 12 = 0}

So, x² - 8x + 12 = 0

x² - 6x 2x + 12 = 0

x (x- 6) 2 (x -6) = 0

(x -6) (x- 2) = 0

x = 6, 2.

So, A = {2, 6) B = {2, 4, 6} C = {2, 4, 6, 8, ….}

D = {6}

D⊂ A ⊂B ⊂C

i e, D ⊂A, D⊂ B, D⊂ C, A⊂ B, A ⊂C and B⊂ C.