Identify whether the size of the set is finite, infinite andcountable, or infini

Question

Identify whether the size of the set is finite, infinite andcountable, or infinite and uncountable. Explain your answerin each case. If the set is finite, write an expression thatgives the number of items in the set. Can someone please helpexplain? a) The set of all rational numbers between 0 and .001 b) The set of all irrational numbers between 0 and .001 c) The set of all distinct ways of ordering a deck of 52cards Identify whether the size of the set is finite, infinite andcountable, or infinite and uncountable. Explain your answerin each case. If the set is finite, write an expression thatgives the number of items in the set. Can someone please helpexplain? a) The set of all rational numbers between 0 and .001 b) The set of all irrational numbers between 0 and .001 c) The set of all distinct ways of ordering a deck of 52cards

Explanation / Answer

a) The set of all rational numbers between 0 and .001    This set is a countable because it isa subset of the countable set Q, the set of rationalnumbers.

b) The set of all irrational numbers between 0 and .001    This set is uncountable.Because any interval in the Real line is uncountable we have theinterval (0, .001) is uncountable.    By (a) the set of rational numbersin this inteval is countable so its complement, that is theset of all irrational numbers between 0 and .001 isuncountable. (Since union of twu countable set is countable.)
c) The set of all distinct ways of ordering a deck of 52cards   It is a finite set. Because there areonly 52! ways of ordering a deck of 52 cards.   ( To place the first card 52 choices, second card 51choices, so on. Hence the number of distinct ways of ordering a deck of 52cards is 52.51.50. ... .2.1 = 52! ) a) The set of all rational numbers between 0 and .001    This set is a countable because it isa subset of the countable set Q, the set of rationalnumbers.

b) The set of all irrational numbers between 0 and .001    This set is uncountable.Because any interval in the Real line is uncountable we have theinterval (0, .001) is uncountable.    By (a) the set of rational numbersin this inteval is countable so its complement, that is theset of all irrational numbers between 0 and .001 isuncountable. (Since union of twu countable set is countable.)
c) The set of all distinct ways of ordering a deck of 52cards   It is a finite set. Because there areonly 52! ways of ordering a deck of 52 cards.   ( To place the first card 52 choices, second card 51choices, so on. Hence the number of distinct ways of ordering a deck of 52cards is 52.51.50. ... .2.1 = 52! )

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