1 Marks Question

Question 11 Mark

Using properties of sets, show that: $A \cap (A \cup B) = A$

Answer

We know that if $A \subset B$ then
$A \cap B = A$
Also $A \subset A \cup B$
$\therefore A \cap (A \cup B) = A$

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Question 21 Mark

Using properties of set, show that: $A \cup (A \cap B) = A$

Answer

We know that if $A \subset B$ then
$A \cap B = B$
Also $A \cap B \subset A$
$\therefore A \cup (A \cap B) = A$

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Question 31 Mark

Show that if $A \subset B$ then $C - B \subset C - A.$

Answer

Let $x \in C - B \Rightarrow x \in C$ and $x \notin B$
$\Rightarrow x \in C$ and $x \notin A$ $[\because A \subset B]$
$\Rightarrow x \in C - A$ Hence $C - B \subset C - A.$

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Question 41 Mark

Show that $A \cap B = A \cap C$ need not imply B = C.

Answer

Let A = {1, 2, 3, 4} , B = {2, 3, 4, 5, 6}, C = {2, 3, 4, 9, 10}
$\therefore A \cap B$= {1, 2, 3, 4} $ \cap $ {2, 3, 4, 5, 6}
= {2,3, 4}
$A \cap C$= {1, 2, 3, 4}, B = {2, 3, 4, 5, 6}, C = {2, 3, 4, 9, 10}
= {2, 3, 4}
$A \cap C$= {1, 2, 3, 4} $ \cap $ {2, 3, 4, 9, 10}
= {2, 3, 4}
Now we have $A \cap B = A \cap C$
But $B \ne C$

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Question 51 Mark

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60° what is ${A'}$?

Answer

Here U = {x : x is a triangle}
A = {x : x is a triangle and has at least one angle different from 60°}
$\therefore A' = U - A = ${x : x is a triangle} - {x : x is a triangle and has atleast one angle different from 60°}
= {x : x is a triangle and has all angles equal to 60°}
= Set of all equilateral triangles.

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Question 61 Mark

Draw appropriate Venn diagram for: A' $\cup$ B'

Answer

The Venn diagram for A' $\cup$ B' The shaded portion represents A' $\cup$ B'

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Question 71 Mark

Draw appropriate Venn diagram for: $(A \cap B)'$

Answer

The Venn diagram for $(A \cap B)'$The shaded portion represents $(A \cap B)'$

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Question 81 Mark

Draw appropriate Venn diagram for: A' $\cap$ B'

Answer

The Venn diagram for A' $\cap$ B' The shaded portion represents A' $\cap$ B'

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Question 91 Mark

Draw appropriate Venn diagram for: $(A \cup B)'$

Answer

The Venn diagram for $(A \cup B)'$The shaded portion represents $(A \cup B)'$

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Question 101 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: {x : 2x + 5 = 9}

Answer

Here $U = \{ x:x \in N\}$
Let A = {x : 2x + 5 = 9{ = {2}
$A' = U - A = \{ x:x \in N\} - \{ 2\}$
$= \{ x:x \in N,x \ne 2\}$

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Question 111 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: {x : x + 5 = 8}

Answer

Here $U = \{ x:x \in N\}$
Let A = {x : x + 5 = 8} = {3}
$A' = U - A = \{ x:x \in N\} -$
$= \{ x:x \in N,x \ne 3\}$

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Question 121 Mark

Taking the set of natural numbesrs as the universal set, write down the complement of the set: {x : x is a perfect cube}

Answer

Here $U = \{ x:x \in N\}$
Let A = {x : x is a perfect cube}
$A' = U - A = \{ x:x \in N\} -${x : x is a perfect cube}
= {x : x $ \in $N , x is not a perfect cube}

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Question 131 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: {x : x is a perfect square}

Answer

Here $U = \{ x:x \in N\}$
Let A = {x : x is a perfect square}
$A' = U - A = \{ x:x \in N\} -$ {x : x is a perfect square}
= { $x:x \in N$, x is not a perfect square}

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Question 141 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: {x : x is a natural number divisible by 3 and 5}

Answer

Here $U = \{ x:x \in N\}$
Let A = {x : x is a natural number divisible by 3 and 5}
$A' = U - A = \{ x:x \in N\} -$ {x : x is a natural number divisible by 3 and 5}
= {$x:x \in N$} - {x : x is a natural number divisible by 15}
= { $x:x \in N$, x is not divisible by 15}

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Question 151 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: {x : x is a prime number}

Answer

Here $U = \{ x:x \in N\}$
Let A = {x : x is a prime number}
$A' = U - A = \{ x:x \in N\} -${x : x is a prime number}
= { $x:x \in N$, x is not a prime number}
or {x : x is positive composite number and x = 1}

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Question 161 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: {x : x is a positive multiple of 3}

Answer

Here $U = \{ x:x \in N\}$
Let A = {x : x is a positive multiple of 3}
$\therefore A' = U - A = \{ x:x \in N\} -${x :x is a positive multiple of 3}
= { $x:x \in N$, x is not a multiple of 3}

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Question 171 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: {x : x is an odd natural number}

Answer

Here $U = \{ x:x \in N\}$

Let A = {x : x is an odd natural number}
$A' = U - A = \{ x:x \in N\} -${x : x is an odd natural number}
= {x : x is an even natural number}

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Question 181 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: $\{ x:x \in N\,\,and\,\,2x + 1 > 10\}$

Answer

Here $U = \{ x:x \in N\}$
Let A = {$x:x \in N$ and 2x + 1 > 10} = {5, 6, 7, 8 , . . . }
$A' = U - A = \{ x:x \in N\} - \{ 5,6,7,8,.....\}$
= {1, 2, 3, 4}

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Question 191 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: $\{ x:x \ge 7\}$

Answer

Here $U = \{ x:x \in N\}$
Let $A = \{ x:x \geqslant 7\} = \{ 7,8,9,10,......\}$
$A' = U - A = \{ x:x \in N\} - \{ 7,8,9,10,.....\}$
= {1, 2, 3, 4, 5, 6}
= {$x:x \in N$ and x < 7}

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Question 201 Mark

Taking the set of natural numbers as the universal set, write down the complement of the set: {x : x is an even natural number}

Answer

Here $U = \{ x:x \in N\}$
Let A = {x : x is an even natural number}
$A' = U - A = \{ x:x \in N\} -${x : x is an even natural number}
= {x : x is an odd natural number}

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Question 211 Mark

If U = {a, b, c, d, e, f, g, h}, find the complement of the set: D = {f, g, h, a}

Answer

$D' = U - D = \{ a,b,c,d,e,f,g,h\} - \{ f,g,h,a\}$={b, c, d, e}

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Question 221 Mark

If U = {a, b, c, d, e, f, g, h}, find the complement of the set: C = {a, c, e, g}

Answer

$C' = U - C = \{ a,b,c,d,e,f,g,h\} - \{ a,c,e,g\}$= {b, d, f, h}

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Question 231 Mark

If U = {a, b, c, d, e, f, g, h}, find the complement of the set: B = {d, e, f, g}

Answer

$B' = U - B = \{ a,b,c,d,e,f,g,h\} - \{ d,e,f,g\}$={a, b, c, h}

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Question 241 Mark

If U = {a, b, c, d, e, f, g, h}, find the complement of the set: A = {a, b, c}

Answer

$A' = U - A = \{ a,b,c,d,e,f,g,h\} - \{ a,b,c\}$= {d, e, f, g, h}

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Question 251 Mark

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2,4, 6, 8} and C = {3, 4, 5, 6}. Find: (B - C)'

Answer

Here U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
B - C = {2, 4, 6, 8} - {3, 4, 5, 6}
= {2, 8}
(B - C)' = U - (B - C) = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 8}
= {1, 3, 4, 5, 6, 7, 9}

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Question 261 Mark

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find: (A')'

Answer

Here U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
A' = U - A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {5, 6, 7, 8, 9}
= {5, 6, 7, 8, 9}
A' = U - A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {5, 6, 7, 8, 9}
= {1, 2, 3, 4} = A

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Question 271 Mark

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2,4, 6, 8} and C = {3, 4, 5, 6}. Find: $(A \cup B)'$

Answer

Here U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
$A \cup B $ = {1, 2, 3, 4} $ \cup $ {2, 4, 6 ,8}
= {1, 2, 3, 4, 6, 8}
$(A \cup B)' = U - (A \cup B)$ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 6, 8}
= {5, 7, 9}

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Question 281 Mark

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2,4, 6, 8} and C = {3, 4, 5, 6}.
Find: $(A \cup C)'$

Answer

Here U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
$A \cup C$ = {1, 2, 3, 4} $ \cap $ {3, 4, 5, 6}
= {1, 2, 3, 4, 5, 6}
$(A \cup C)' = U - (A \cup C)$= {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 5, 6}
={7, 8, 9}

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Question 291 Mark

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2,4, 6, 8} and C = {3, 4, 5, 6}. Find: B'

Answer

Here U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}
B' = U - B = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}
= {1, 3, 5, 7, 9}

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Question 301 Mark

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2,4, 6, 8} and C = {3, 4, 5, 6}. Find : $\begin{equation} \mathbf{A}^{\prime} \end{equation}$

Answer

we have to find the complement of, which is given by (U - A)
Where U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4},
$\therefore$ $\begin{equation} \mathbf{A}^{\prime} \end{equation}$ = U - A
$\Rightarrow$ $\begin{equation} \mathbf{A}^{\prime} \end{equation}$ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4}
$\Rightarrow$ $\begin{equation} \mathbf{A}^{\prime} \end{equation}$ ={5, 6, 7, 8, 9}

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Question 311 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: C - B

Answer

Here A = {3, 6,9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}

C - B = {2,4,6,8,10,12,14,16} - {4,8,12,16,20}
= {2,6,10,14}

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Question 321 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}. Find: B - D.

Answer

Here A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}
B - D = {4, 8, 12, 16, 20} - {5, 10, 15, 20}
= {4, 8, 12, 16}

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Question 331 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}. Find: B - C

Answer

Here A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}
C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}
B - C = {4, 8, 12, 16, 20} - {2, 4, 6, 8, 10, 12, 14, 16}
= {20}

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Question 341 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}. Find: D - A

Answer

Here A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}
D - A = {5, 10, 15, 20} - {3, 6, 9, 12, 15, 18, 21}
= {5, 10, 20}

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Question 351 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: C - A

Answer

Here A = {3, 6,9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}

C - A = {2,4,6,8,10,12,14,16} - {3,6,9,12,15,18,21}
= {2,4,8,10,14,16}

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Question 361 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: B - A

Answer

Here A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}
B - A = {4, 8, 12, 16, 20} - {3, 6, 9, 12, 15, 18, 21}
= {4, 8, 16, 20}

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Question 371 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: A - D

Answer

Here A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}
A - D = {3, 6, 9, 12, 15, 18, 21} - {5, 10, 15, 20}
= {3, 6, 9, 12, 18, 21}

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Question 381 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: A - C

Answer

Here A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}
C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}
A - C = {3, 6, 9, 12, 15, 18, 21} - {2, 4, 6, 8, 10, 12, 14, 16}
={3, 9, 15, 18, 21}

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Question 391 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: D - C

Answer

Here A = {3, 6,9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}
D - C = {5,10,15,20} - {2,4,6,8,10,12,14,16}
= {5,15,20}

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Question 401 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: C - D

Answer

Here A = {3, 6,9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}

C - D = {2,4,6,8,10,12,14,16} - {5,10,15,20}
= {2,4,6,8,12,14,16}

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Question 411 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: D - B

Answer

Here A = {3, 6,9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}
C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}

D - B = {5,10,15,20} - {4,8,12,16,20}
= {5,10,15}

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Question 421 Mark

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find: A - B

Answer

Here A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}
C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}
A - B = {3, 6, 9, 12, 15, 18, 21} - {4, 8, 12, 16, 20}
= { 3, 6, 9, 15, 18, 21}

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Question 431 Mark

Given pair of sets are disjoint? {x : x is an even integer} and {x : x is an odd integer}

Answer

Let A = {x : x is an even integer}
and B = {x : x is an odd integer}
$\therefore A \cap B = \phi $
Hence A and B are disjoint.

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Question 441 Mark

Given pair of sets are disjoint? {a, e, i, o, u} and {c, d, e, f}

Answer

(ii) Let A = {a, e, i, o, u}
and B = {c, d, e, f}
$\therefore A \cap B ≠ \phi$
Hence A and B are not disjoint.

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Question 451 Mark

Given pair of sets are disjoint? {1, 2, 3, 4} and {x : x is a natural number and $4 \leq x \leq 6$

Answer

Let A = {1, 2, 3, 4}
and B = {x : x is a natural number and $4 \leq x \leq 6$
= {4, 5, 6}
$\therefore A \cap B = \{ 1,2,3,4\} \cap \{ 4,5,6\}$
= {4}
Hence A and B are not disjoint.

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Question 461 Mark

If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number} and D = {x : x is a prime number}, find: $C \cap D$

Answer

Here A = {x : x is a natural number} = {1, 2, 3, 4, 5, . . .}
B = { x : x is an even natural number} = {2, 4, 6, . . . }
C = {x : x is an odd natural number} = {1, 3, 5, 7, . . .}
and D = {x : x is a prime nmber} = {2, 3, 5, 7. . . . }

$C \cap D$ = {x : x is an odd natural number} $ \cap ${x : x is a prime number}
= {x : x is an odd prime number}

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Question 471 Mark

If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number} and D = {x : x is a prime number}, find: $B \cap D$

Answer

$B \cap D$ = {x : x is an even natural number) $ \cap ${x : x is a prime number}
= {2}

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Question 481 Mark

If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number} and D = {x : x is a prime number}, find: $B \cap C$

Answer

$B \cap C$= {x : x is an even natural number} $ \cap${x : x is an odd natural number}
$ = \phi $

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Question 491 Mark

If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number} and D = {x : x is a prime number}, find: $A \cap D$

Answer

$A \cap D$= {x : x is a natural number) $ \cap${x : x is a prime number}
= {x : x is a prime number}
= D

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Question 501 Mark

If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number} and D = {x : x is a prime number}, find: $A \cap C$

Answer

$A \cap C$ = {x : x is a natural number} $ \cap ${x : x is an odd natural number}
= { x : x is an odd natural number }
= C

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