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# System prepended metadata

title: DM _CH(2)

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<p style="font-size: 14px;text-align:center">
The Islamic University of Gaza
</p>

<p style="font-size: 14px;text-align:center">
    Discrete Mathematics Lab - 2023/2024
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<p style="color:#000033;font-weight: bold;text-align:center">
   Discussion || ch(2)
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<p style="text-align:right">
    Eng: Amal I. Mahfouz , Eng: hashem hjazy
</p>



****

<p style="color:#000066;font-weight: bold;">
   section 2.1
</p>

<p style="color:#000066;font-weight: bold;">
  *  2.1.2:  Use set builder notation to give a description of each of these sets.
    
 a) {0,3,6,9,12}
    
       { X ∣ X = 3n, n ∈ Z, 0 ≤ n ≤ 4}
       
    
 b) {−3,−2,−1,0,1,2,3}
    
       { X | -3 <= X <= 3, x ∈ Z }
 c) {m,n,o,p}
    
       {X ∣ X ∈ {m, n, o, p } }
</p>


<p style="color:#000066;font-weight: bold;">
  *2.1.7: Determine whether each of these pairs of sets are equal.
    
    Note  : sets do not contain duplicate elements.
    
 a) {1,3,3,3,5,5,5,5,5},{5,3,1}
    
    Equal
    
 b) {{1}},{1,{1}}
    
     Not Equal, the first set has one element while the second set has  two elements. 
    
 c) ∅,{∅}
    
    Not Equal, the first set has no elements, the second set has one 
element (∅) 
</p>

<p style="color:#000066;font-weight: bold;">
  *2.1.9:  For each of the following sets, determine whether 2 is an element of that set.
    
 a)  x ∈ R|xisaninteger greater than 1 
    
        2 is an element of this set.
        ex :  set={2,3,4,5,etc.}
    
 b)  x ∈ R|xisthe square of an integer 
    
       2 is not an element of this set.
       ex :  set={0,1,4,9,16,etc.}
    
    
 c) {2,{2}}
    
       2 is an element of this set.
       ex : This set contains two elements: the number 2 and the set {2}
    
    
 d) {{2},{{2}}}
    
       2 is not an element of this set.
       ex: The number 2 itself is not directly listed as an element of this set
    
 e) {{2},{2,{2}}}
        
       2 is not an element of this set.
    
 f) {{{2}}}
    
       2 is not an element of this set.
</p>

<p style="color:#000066;font-weight: bold;">
   *2.1.11:  Determine whether each of these statements is true or false.
    
 a) 0 ∈∅    
    
    False (0 is not an element of the empty set.)
    
 b) ∅ ∈ {0}
    
    False (The set {0} only contains 0, not the empty set ∅.)
    
    
 c) {0} ⊂ ∅
    
     False
    
 d) ∅ ⊂ {0}
    
     True  (The empty set ∅ is a subset of every set)
    
 e) {0} ∈ {0}
    
     False (the set {0} is an element of the set {0}? No , 
            because the latter is contain element 0 not element {0})
    
 f) {0} ⊂ {0}
    
    False
    
 g) {∅} ⊆ {∅}
    
    True
 
 
 
</p>

<p style="color:#000066;font-weight: bold;">
   *2.1.21: What is the cardinality of each of these sets?
    
 a) {a}
    
    1
    
 b) {{a}}
     
    1
    
 c) {a,{a}}
    
    2
    
 d) {a,{a},{a,{a}}}
    
    3
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</p> 


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<p style="color:#000066;font-weight: bold;">
   section 2.2
</p>

<p style="color:#000066;font-weight: bold;">
   *2.2.3:  Let A ={1,2,3,4,5} and B ={0,3,6}.Find
    
    
 a) A∪B.
    
    A ∪ B = {1,2,3,4,5} ∪ {0,3,6} = {0,1,2,3,4,5,6}
 b) A∩B.
    
    A ∩ B = {1,2,3,4,5} ∩ {0,3,6} = {3}
    
 c) A−B.
    
    A − B = {1,2,3,4,5} − {0,3,6} = {1,2,4,5}
    
 d) B−A.
    
    B − A= {0,3,6} − {1,2,3,4,5} = {0,6}
</p>
 
 <p style="color:#000066;font-weight: bold;">
  *2.2.7: Prove the domination laws in Table 1 by showing that: 
    
    to solve this question , let us remember : 
    Universal Set (U): Contains all possible elements.
    Empty Set (∅): Contains no elements.
    Set A: Any arbitrary set.
    
 a) A ∪ U = U.
    
        the universal set contains all elements , so that the union of any set 
        with the universal set is the universal set 
        if x ∈ A , then x is in A ∪ U ; because A ⊆ U
       
 b) A ∩ ∅ = ∅.
    
    the intersection of any set with the empty set is the empty set ;
    because there are no common elements in ∅.
</p>

<p style="color:#000066;font-weight: bold;">
   *2.2.14:Find the sets A and B if A−B ={1,5,7,8}, B−A =
 {2, 10},and A ∩ B ={3,6,9}. 
     
    A = {1,5,7,8,3,6,9}
    B = {2,10,3,6,9}
    
    
</p>

<p style="color:#000066;font-weight: bold;">
   2.2.21: Show that if A and B are sets, then
    
 a) A − B = A ∩ 'B.
    
    A − B = { x:x ∈ A and x (not ∈) B}. the set of elements that are in  𝐴 but not in 𝐵
    A ∩ 'B = {x:x ∈ A and x (not ∈ B)}.  the set of elements that are in both 𝐴 and 
    complement 𝐵
    
    Since both sets contain the same elements (those that are in  𝐴 but not in 𝐵) ; 
    Thus, the statement is true:  A - B = A ∩ 'B

    
    these two sets are not the same,  𝐴 − 𝐵 contains elements in 𝐴 but not in 𝐵,
    while 𝐴 ∩ 𝐵 contains elements that are in both 𝐴 and  𝐵. 
    These are two different operations, so 𝐴 − 𝐵 ≠ 𝐴 ∩𝐵


    
 b) (A ∩ B) ∪(A ∩ 'B) = A.
    
    
    A ∩ 'B = {x:x∈A and x (not ∈)B} the set of elements that are in both 𝐴 and 
    complement 𝐵
    A ∩ B = {x:x ∈ A and x ∈ B}.  the set of elements that are in both 𝐴 and 𝐵
    
    The union of these two sets,(A ∩ B) ∪(A ∩ 'B) includes all the elements that are 
    either in both A and 𝐵, or in  A but not in 𝐵.
    
    This means all the elements that are in 𝐴 in either case are included in the union.
    
    
</p>