HELP

8)  U  =  {1, 2, 3, 4, 5, 6}     A  =  {2, 3, 6}

c)  A'  =  {1, 4, 5}     --->     A ^A'  =  {2, 3, 6} ^ {1, 4, 5}  =  { }      (the null set)

d)  U'  =  { }     --->     U' ^ A  =  { } ^ {2, 3, 6}  =  { }

9)  P  =  {1, 3, 5, 7}     Q  =  {3, 7, 9, 11}     R  =  {1, 5, 8, 11}

a)   Show that:  P v Q  =  Q V P

      --->    Since  P v Q  =  {1, 3, 5, 7} v {3, 7, 9, 11}  =  {1, 3, 4, 7, 9, 11}

      --->     and   Q v P  =  {3, 7, 9, 11} v {1, 3, 5,7 }  =  {1, 3, 4, 7, 9, 11},

                therefore  P v Q  =  Q v P

b)  Show that:  (P v Q) v R  =  P v (Q v R)

     --->  Since (P v Q) v R  =  ( {1, 3, 5, 7} v {3, 7, 9, 11} ) v {1, 5, 8, 11}  

                                         =  {1, 3, 5, 7, 9, 11} v {1, 5, 8, 11}  =  {1, 3, 5, 7, 8, 9, 11}

     --->  and  P v (Q v R)  =  {1, 3, 5, 7} v ( {3, 7, 9, 11} v {1, 5, 8, 11} )

                                        =  {1, 3, 5, 7} v {1, 3, 5, 7, 8, 9, 11}  =  {1, 3, 5, 7, 8, 9, 11}

             therefore (P v Q) v R  =  P v (Q v R)

c)  Show that:  P ^ Q  =  Q ^ P

     --->  Since P ^ Q  =  {1, 3, 5, 7} ^ {3, 7, 9, 11}  =  {3, 7}

     --->   and  Q ^ P  =  {3, 7, 9, 11| ^ {1, 3, 5, 7}  =  {3, 7}

             therefore  P ^ Q  =  Q ^ P

d)  Show that:  (P ^ Q) ^ R  =  P ^ (Q ^ R)

     --->  Since  (P ^ Q) ^ R  =  ( {1, 3, 5, 7} ^ {3, 7, 9, 11} ) ^ {1, 5, 8, 11}

                                           =  {3, 7} ^ {1, 5, 8, 11}  =  { }

             and  P ^ (Q ^ R)  =  {1, 3, 5, 7} ^ ( {3, 7, 9, 11} ^ {1, 5, 8, 11} )

                                        =  {1, 3, 5, 7} ^ { }  =  { }

             therefore  (P ^ Q) ^ R  =  P ^ (Q ^ R)

e)  Show that  P v (Q ^ R)  =  (P v Q) ^ (P v R)

     --->  Since  P v (Q ^ R)  =  {1, 3, 5, 7} v ( {3, 7, 9, 11} ^ {1, 5, 8, 11} )

                                          =  {1, 3, 5, 7} v { 11  }  =  {1, 3, 5, 7, 11}

             and  (P v Q) ^ (P v R)  =  ( {1, 3, 5, 7} v {3, 7, 9, 11} ) ^ ( {  1, 3, 5, 7} v {1, 5, 8, 11} )

                                                 =  {1, 3, 5, 7, 9, 11} ^ {1, 3, 5, 7, 8, 11}  =  {1, 3, 5, 7, 11}

             therefore  P v (Q ^ R)  =  (P v Q) ^ (P v R)

f)  Show that  P ^ (Q v R)  =  (P ^ Q) v (P ^ R)

     --->  Since  P ^ (Q v R)  =  {1, 3, 5, 7} ^ ( {3, 7. 9. 11} v {1, 5, 8, 11} )

                                           =  {1, 3, 5, 7} ^ {1, 3, 5, 7, 8, 9, 11}  =  {1, 3, 5, 7 }

             and  (P ^ Q) v (P ^ R)  =  ( {1, 3, 5, 7} ^ (3, 7, 9, 11} ) v ( {1, 3, 5, 7} ^ {1, 5, 8, 11} )

                                                 =  {3, 7} v {1, 5}  =  {1, 3, 5, 7}

Hi Gino,

what does A' mean?

U={1,2,3,4,5,6}        A={2,3,6}

You said it was A'={1,4,5} but

Why does it have anything to do with U ?

I thought it might be the set of everything that is not in A but that would include an infinite number of elements.

I thought that maybe

\(U \cap A' = \{1,4,5\}\)

Could you please explain ?     

Hi again Gino,

I've just been doing some homework

https://www.mathsisfun.com/sets/symbols.html

ok So U is the universal set.   NOW I understand your answer. thanks.      

I think these symbols all mean "compliment of A"

\(A', \qquad A^c, \qquad \bar A\)

I am only familiar with the last one. ://