Question

- Assume that the following variables are set as follow

P is True, Q is False, R is False.

- Solve for X

X = (P ∧ ~Q) ∨ (~P ∧ ~R)

- Solve for Y

Y=(P Q ) ∨ (R ~P)

20 points

- Give one example and the mathematical symbol for the following:

- A universal set
- A subset
- A proper subset
- An empty set
- The intersect of two sets.

25 points

- Convert the following the numbers (must show all work)

- 241 decimal to Binary
- 11111101 to Binary
- 11101010 to Hex
- 73 Octal to Binary
- 2AB Hex to Binary

25 points

- Perform the following arithmetic operation (MUST SHOW ALL WORK):

- X = 11101 + 1010 + 11011
- Y = 1100 – 111

10 Points

- Solve the following and SHOW HOW DID YOU REACH your answer

- If P is False, Q is unknown

P Q

- P is True, Q is unknown

P ∧ Q

10 points

6. perform the following using 2s complement

X = - 87 – 32

Y = 89 -121

10 Points

Answer #1

Ans: (a)

**X = (P ∧ ~Q) ∨ (~P ∧
~R)**

Given That:: P=True, Q= False, R=False

= (P ∧ ~Q ∨ ~P) ∧ (P ∧
~Q ∨ ~R) **Note: ([P V
~P]=True)**

= (True ∧ ~Q) ∧ (P ∧ ~Q ∨ ~R) [ ~Q=true ;because Q is false] [ ~R=true ;because is false]

=(True )∧ (P ∧ ~Q ∨ ~R)

= (True ∧ P) ∧(True ∧ ~Q ) ∨(True∧~R)

= (True) ∧ (True) V (True)

**Ans:
X=True**

**Y=(P Q ) ∨ (R
~P)**

** = (~P ∨Q)
∨** ((~R**∨~P)**∧ (**~(~P)∨R)) Note: [ P Q=~P ∨Q and (R
~P)=**(~R**∨~P)**∧ (**~(~P)∨R)**

** =**(~P ∨Q) ∨ ((~R∨~P)∧ (~P∨R))

=(~P ∨Q ∨ ~R∨~P) ∧(~P ∨Q ∨ ~P∨R))

=(True ∨ Q ∨ ~R) ∧(True
∨Q ∨R) **[True∨ A=True
A=true/false]**

** =** True ∧ True

**Ans
:Y=True**

**Que
:** **Give one example and the mathematical
symbol for the following:**

**ANS:**

**A universal
set**:

**Symbol:
U**

**Example: 1.**
Set of Integer
U={1,2,3,4,5,6,7,8,9,10}

** 2.**
Set of Humanbeing U= {Male,
Female}

**subset**

**Symbol:
⊂**

**Example**: E={Set of even number} **E⊂U =**{2,4,6,8,10} (E is a subset of U)

O={Set
of odd number} **O⊂U=**{1,3,5,7,9} (O is a subset of U)

**A proper subset:**

Example: A={2,4}

B={1,2,3,4}

A
⊂ B={1,4} [A is a proper set of B : A ⊂ B and **A ≠ B** ]

**An empty set**

**Symbol:** ∅

A=:{1,3,5}

B={2,4,6}

A⊂ B={∅}*

Example: N={ A set with 0 element ) or N={}

**The intersect of two
sets:**

**Symbol:** ∩

**Example**
P={1,2,3,4,5}

Q={2,4,6,8}

**P ∩ Q={2,4} {common element
between two set}**

**Que: Convert the following the numbers
(must show all work)**

ANS:

a. **241
decimal to Binary**

241 divide 2= 120; r1=1 [ Remainder]

120 divide 2=60; r2=0

60 divide 2=30; r3=0

30 divide 2=15; r4=0

15 divide 2=7; r5=1

7 divide 2=3; r6=1

3 divide 2=1; r7=1

1 divide 2=0; r8=1=1 [ stop when we get division 0]

(r8,r7,r6,r5,r4,r3,r2,r1)=(11110001)_{2}

**Ans****:(**
**241)=(11110001)**_{2}

**B.
11111101 to Binary**

Step:
(1x2^{7})+(1x2^{6})+(1x2^{5})+(1x2^{4})+(1x2^{3})+(1x2^{2})+(0x2^{1})+(1x2^{0})

= 128+64+32+16+8+4+0+1

=253

**Ans
(11111101)**_{2}**=
(253)**_{10}

**C.
11101010 to Hex**

Ans:
(11101010)_{2}=()_{2}^{4}

(1110)(1010)=(**(1x2**^{3}**)+(1x2**^{2}**)+(0x2**^{1}**)+(1x2**^{0}**))
((1x2**^{3}**)+(0x2**^{2}**)+(1x2**^{1}**)+(0x2**^{0}**))**

** =
(8+4+0+1)(8+0+2+0)**

** =(13)(10)
[ In Hexadecimal: 13= D; 10=A**

** =(DA)**_{16}

**ANS****:**
**11101010 to Hex
=(DA)**_{16}

**D. 73 Octal to
Binary:**

(73)_{8=}(7)_{2}^{3}(3)_{2}^{3}

=
(101)(011) [ convert (7)_{2=}(101); (3)_{2}=(011)

=(101011)_{2}

_{ }**Ans:
73 Octal to Binary=(101011)**_{2}

**E. 2AB Hex to
Binary**

value of A is in decimal=10 and B is 11

(2AB)_{16}=()_{2}^{4}

=(0010)_{2}
(1010)_{2} (1011)_{2} [Note: 2 to binary=0010 (4 digit because binary 4
digit equal 1 digit of hexadeciamal); A=1010; B=1011]

=(0
0 1 0 1 0 1 0 1 0 1 1)_{2}

_{ }_{ }**Ans:
2AB Hex to Binary =(0 0 1 0 1 0 1 0 1 0 1
1)**_{2}

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