Question

1. Assume that the following variables are set as follow

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

1. Solve for X

X = (P ∧ ~Q) ∨ (~P ∧ ~R)

2. Solve for Y

Y=(P  Q ) ∨ (R  ~P)

                                                                                                                                                                           20 points

2. Give one example and the mathematical symbol for the following:

1. A universal set
2. A subset
3. A proper subset
4. An empty set
5. The intersect of two sets.

25 points

3. Convert the following the numbers (must show all work)

1. 241 decimal to Binary
2. 11111101 to Binary
3. 11101010 to Hex
4. 73 Octal to Binary
5. 2AB Hex to Binary

25 points

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

1. X = 11101 + 1010 + 11011
2. Y = 1100 – 111

10 Points

5. Solve the following and SHOW HOW DID YOU REACH your answer

1. If P is False, Q is unknown

     P  Q

2. 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: (1x2⁷)+(1x2⁶)+(1x2⁵)+(1x2⁴)+(1x2³)+(1x2²)+(0x2¹)+(1x2⁰)

       = 128+64+32+16+8+4+0+1

       =253

                                            Ans (11111101)₂= (253)₁₀

C. 11101010 to Hex

Ans: (11101010)₂=()₂⁴

(1110)(1010)=((1x2³)+(1x2²)+(0x2¹)+(1x2⁰)) ((1x2³)+(0x2²)+(1x2¹)+(0x2⁰))

                    = (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)₂³(3)₂³

         = (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=()₂⁴

            =(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