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Mathematics LibreTexts

Exercises

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  • Page ID
    4911
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    Exercise \(\PageIndex{1}\): Set Operations

    Let \(A = \{1, 5, 31, 56, 101\}\), \(B = \{22, 56, 5, 103, 87\}\), \(C = 41, 13, 7, 101, 48\}\), and \(D = \{1, 3, 5, 7...\}\)

    Give the sets resulting from:

    1. \(A \cap B\)
    2. \(C \cup A\)
    3. \(C \cap D\)
    4. \((A \cup B) \cup (C \cup D)\) 

    Exercise \(\PageIndex{2}\): True or False

    1. \(7 \in \{6, 7, 8, 9\}\)
    2. \(5 \notin \{6, 7, 8, 9\}\)
    3. \(\{2\} \nsubseteq \{1, 2\}\)
    4. \(\emptyset \nsubseteq \{\alpha, \beta, x\}\)
    5. \(\emptyset = \{\emptyset\}\)

    Exercise \(\PageIndex{3}\): Subsets

    List all the subsets of:

    1. \(\{1, 2, 3\}\)
    2. \(\{\phi, \lambda, \Delta, \mu\}\)
    3. \(\{\emptyset\}\)

    Exercise \(\PageIndex{4}\): Venn Diagram

    A survey of 100 university students found the following data on their food preferences:

    • 54 preferred Italian cuisine
    • 29 preferred Asian-style cooking
    • 16 preferred both Italian and Asian-style foods
    • 19 preferred both Asian-style and Indian dishes
    • 10 preferred both Italian and Indian cuisines
    • 5 liked them all
    • 11 did not like any of the options

    How many students preferred:

    1. Only Indian food?
    2. Only Italian food?
    3. Only one food?

    Exercise \(\PageIndex{5}\): Symbols

    Assume that the universal set is the set of all integers. 
    Let
     \(A=\{-7,-5,-3,-1,1,3,5,7\} \)
     \(B =\{ x \in {\bf Z}| x^2 <9 \} \)  
      \(C= \{2,3,4,5,6\}\)
      \(D=\{x \in {\bf Z}| x \leq 9 \}\)

    In each of the following fill in the blank with most appropriate  symbol from  \(\in, \notin, \subset, =,\neq,\subseteq\), so that resulting statement is true.

    A-----D
    3-----B
    9-----D

    {2}-----\(C^c\)
    \(\emptyset\)-----D
    A-----C
    B-----C
    C-----D
     0-----\(A \cap D\)
     0-----\(A \cup  D\)

    Exercise \(\PageIndex{6}\):

    Given subsets \(A,B,C\) of a universal set \(U\), prove the statements that are true and give counter examples to disprove those that are false.

    1.  \( A-(B \cap C)=(A-B) \cup(A-C).\)
    2. If \( A \cap B= A \cap C\) then \(B= C\).
    3. If \( A \cup B= A \cup C\) then \(B= C\).
    4.  \( A-(B - C)=(A-B)-C.\)
    5.   If \(A \times B \subseteq  C \times D\) then \(A\subseteq C\) and \( B \subseteq D.\)
    6.  If \(A\subseteq C\) and \( B \subseteq D\) then \(A \times B \subseteq  C \times D.\)

    Exercise \(\PageIndex{7}\):

    Let \(A = \{ r,e,a,s,o,n,i,g\}, B = \{m,a,t,h,e,t,i,c,l\} \) and \( C \) = the set of vowels.  Calculate:

    1.  \(A \cup B \cup C.\)
    2.  \(A \cap B.\)
    3. \({C}^c\).

    Exercise \(\PageIndex{8}\):

    Given subsets \(A,B,C\) of a universal set \(U\), prove the statements that are true and give counter examples to disprove those that are false.

    1. \(P(A \cup B) = P(A) \cup P(B).\)
    2. \(P(A \cap B) = P(A) \cap P(B).\)
    3. \(P(A^c)=(P(A))^c\)
    4. \(P(A - B) = P(A) - P(B).\)

    Exercise \(\PageIndex{9}\):

    Consider the following sets:

    \(A=\{x \in {\bf Z}| x= 2m, m \in {\bf Z}\} \) and \(B=\{x \in {\bf Z}| x= 2(n-1), n \in {\bf Z}\} \).

    Are \(A\) and \(B\) equal? Justify your answer.

     

    Exercise \(\PageIndex{10}\):

    Let \(A=\{1,3,5\} \), and 
     \(B =\{ a,b \} \).

    Then 

    1. Find \( A \times B\) and \(B \times A\).
    2. Are \(A \times B\) and \(B \times A\) equal? Justify your answer.