Set Questions

**cloud5** · June 17th 2009, 05:37 AM

Help me solve the question please? They make me headache.

1. Show
i. $(A' \cup B') \cap (A' \cup B) \cap (A \cup B)=A' \cap B$

ii. $(A \cup B') \cap (A \cup B)=A$

iii. $(A \cup B' \cup C') \cap [A \cup (B \cap C)]=A$

iv. $A \cup (A' \cup B)'=A$

2. A certain sixth form science class consists of 90 pupils, all of whom take mathematics, 45 study chemistry, 42 study biology, and 52 study physics. Of those taking three subjects, 18 study physics and chemistry, 22 study physics and biology, and 16 study chemistry and biology while 5 take all four subjects. Find the number of pupils who study mathematics only.
Answer: 2

Note: I'd like a Venn Diagram and the formula for question 2.

**mathaddict** · June 17th 2009, 05:46 AM

Originally Posted by cloud5

Help me solve the question please? They make me headache.

1. Show
i. $(A' \cup B') \cap (A' \cup B) \cap (A \cup B)=A' \cap B$

(A' u B') n [( A' n A) u B ]

=( A' u B' ) n ( nullset u B )

=(A' u B' ) n B

= (A' n B ) u ( B' n B )

= ( A' n B ) u null set

= you got ur prove .

**mathaddict** · June 17th 2009, 05:51 AM

Originally Posted by cloud5

Help me solve the question please? They make me headache.

1. Show

ii. $(A \cup B') \cap (A \cup B)=A$

iii. $(A \cup B' \cup C') \cap [A \cup (B \cap C)]=A$

(2) A u ( B' n B ) = A u null set - distributive

(3) A u [( B' u C' ) n (B n C )] - distributive

= A u [( B n C )' n ( B n C )] -- de morgans

= A u null set

= A

(4) This is not clear . Its A u (A' u B)' or [A u (A' u B)]'

**cloud5** · June 17th 2009, 06:01 AM

Originally Posted by mathaddict

(2) A u ( B' n B ) = A u null set - distributive

(4) This is not clear . Its A u (A' u B)' or [A u (A' u B)]'

Opss... made a mistake there... it is

$A \cup (A' \cup B)'=A$

**mathaddict** · June 17th 2009, 06:11 AM

Originally Posted by cloud5

Opss... made a mistake there... it is

$A \cup (A' \cup B)'=A$

( A n universal set ) u ( A n B' ) = A n ( universal set u B )

= A n universal set

=A

**Soroban** · June 17th 2009, 06:50 AM

Hello, cloud5!

I'll do the first two.

We need a couple of Theorems:

. . $\begin{array}{cc} [1] & P \cap P' \:=\:\emptyset \\ \\[-3mm] [2] & P \cup \emptyset \:=\:P \end{array}$

$(ii)\;\;(A \cup B') \cap (A \cup B) \;=\;A$

$(A \cup B') \cap (A \cup B)$

. . $\begin{array}{ccc}= & A \cup (B' \cap B) & \text{Distr.} \\ \\ = &A \cup \emptyset & [1] \\ \\ =& A & [2] \end{array}$

$(i)\;\;(A' \cup B') \cap (A' \cup B) \cap (A \cup B) \:=\:A' \cap B$

$(A' \cup B') \cap (A' \cup B) \cap (A \cup B)$

. . $\begin{array}{cccc}= & \bigg[A' \cup (B' \cap B)\bigg] \cap (A \cup B) & \text{Distr.} \\ \\ = & \bigg[A' \cup \emptyset\bigg] \cap (A \cup B) & [1] \\ \\ = & A' \cap (A \cup B) & [2] \\ \\ = & (A' \cap A) \cup (A' \cap B) & \text{Distr.} \\ \\ = & \emptyset \cup (A' \cap B) & [1] \end{array}$

. . $\begin{array}{cccc} = & \qquad\qquad A' \cap B & \qquad\qquad\quad\; [2] \end{array}$

Thread: Set Questions

LinkBack

Thread Tools

Display

Set Questions

Similar Math Help Forum Discussions

More log questions

Please someone help me with just 2 questions?

Some Questions !

Plz solve these questions.(Trignometry statement questions)

Assorted algebraic questions ( Read this before asking some questions! )

Search Tags