Question

let sets ( a ) and ( b ) be defined as follows.

( a ) is the set of integers greater than or equal to ( -10 ) and less than or equal to ( -3 )
( b = {-29, -23, 22, 26, 27, 30} )

(a) find the cardinalities of ( a ) and ( b ).

( n(a) = square ) ( n(b) = square )

(b) select true or false.

	true	false
( -16 in a )	( circ )	( circ )

| ( 28
otin b ) | ( circ ) | ( circ ) |

( -9 in a )

( circ )

( circ )

Explanation:

For \(-9\in A\): Set \( A \) has integers up to \(-3\) and from \(-10\) (inclusive). \(-9\) is greater than \(-10\) but \( A \) has integers less than or equal to \(-3\)? Wait, \( A \) is integers greater than or equal to \(-10\) and less than or equal to \(-3\). \(-9\) is greater than \(-10\) but \(-9 > -3\)? Wait, \(-9\) is greater than \(-10\) and \(-9\) is less than \(-3\)? Wait, \(-9\) is between \(-10\) and \(-3\)? Wait, \(-10\leq x\leq -3\). \(-9\) is greater than \(-10\) and less than \(-3\) (since \(-9 < -3\) is false? Wait, \(-9\) is less than \(-3\)? Yes, \(-9 < -3\) (because \(-9\) is more negative). Wait, no: \(-3\) is greater than \(-9\) (since on number line, \(-3\) is to the right of \(-9\)). So \(-9\) is between \(-10\) and \(-3\)? Wait, \(-10\leq -9\leq -3\)? Yes, because \(-9\geq -10\) and \(-9\leq -3\). Wait, but earlier calculation for \( n(A) \): we had from \(-10\) to \(-3\). Let's list \( A \): \(-10,-9,-8,-7,-6,-5,-4,-3\). Oh! I made a mistake earlier. Wait, integers greater than or equal to \(-10\) and less than or equal to \(-3\): so the numbers are \(-10, -9, -8, -7, -6, -5, -4, -3\). So \(-9\) is in \( A \)? Wait, earlier calculation for \( n(A) \): I thought the formula was \( b - a + 1 \), where \( a=-10 \), \( b=-3 \). So \(-3-(-10)+1 = 8 \). Let's list them: \(-10, -9, -8, -7, -6, -5, -4, -3\) (8 elements). So \(-9\) is in \( A \). Wait, so my earlier thought was wrong. So \(-9\) is in \( A \)? Wait, \(-9\) is greater than or equal to \(-10\) (yes, \(-9\geq -10\)) and less than or equal to \(-3\) (yes, \(-9\leq -3\)). So \(-9\in A\) is True? Wait, but let's re - check the set \( A \) definition: "integers greater than or equal to \(-10\) and less than or equal to \(-3\)". So \(-10,-9,-8,-7,-6,-5,-4,-3\): 8 elements. So \(-9\) is in \( A \). So \(-9\in A\) is True? Wait, but earlier when I thought about \(-9\), I made a mistake. So correction: \(-9\) is in \( A \), so \(-9\in A\) is True. Wait, but let's re - calculate \( n(A) \): from \(-10\) to \(-3\) inclusive. The number of integers is \((-3)-(-10)+1=-3 + 10 + 1=8\). So the elements are \(-10,-9,-8,-7,-6,-5,-4,-3\). So \(-9\) is in \( A \). So for \(-9\in A\), it is True. Wait, but earlier I was confused. So that's a mistake. So let's re - do the part (b) for \(-9\in A\):

• For \(-9\in A\): Since \( A=\{-10,-9,-8,-7,-6,-5,-4,-3\} \), \(-9\) is in \( A \), so this is True.

Final Answers

Part (a)

\( n(A)=\boldsymbol{8} \), \( n(B)=\boldsymbol{6} \)

Part (b)

• \(-23\in B\): True
• \(-16\in A\): False
• \(28

otin B\): True

• \(-9\in A\): True

(For the table, mark True for \(-23\in B\), False for \(-16\in A\), True for \(28
otin B\), True for \(-9\in A\))

Answer:

For \(-9\in A\): Set \( A \) has integers up to \(-3\) and from \(-10\) (inclusive). \(-9\) is greater than \(-10\) but \( A \) has integers less than or equal to \(-3\)? Wait, \( A \) is integers greater than or equal to \(-10\) and less than or equal to \(-3\). \(-9\) is greater than \(-10\) but \(-9 > -3\)? Wait, \(-9\) is greater than \(-10\) and \(-9\) is less than \(-3\)? Wait, \(-9\) is between \(-10\) and \(-3\)? Wait, \(-10\leq x\leq -3\). \(-9\) is greater than \(-10\) and less than \(-3\) (since \(-9 < -3\) is false? Wait, \(-9\) is less than \(-3\)? Yes, \(-9 < -3\) (because \(-9\) is more negative). Wait, no: \(-3\) is greater than \(-9\) (since on number line, \(-3\) is to the right of \(-9\)). So \(-9\) is between \(-10\) and \(-3\)? Wait, \(-10\leq -9\leq -3\)? Yes, because \(-9\geq -10\) and \(-9\leq -3\). Wait, but earlier calculation for \( n(A) \): we had from \(-10\) to \(-3\). Let's list \( A \): \(-10,-9,-8,-7,-6,-5,-4,-3\). Oh! I made a mistake earlier. Wait, integers greater than or equal to \(-10\) and less than or equal to \(-3\): so the numbers are \(-10, -9, -8, -7, -6, -5, -4, -3\). So \(-9\) is in \( A \)? Wait, earlier calculation for \( n(A) \): I thought the formula was \( b - a + 1 \), where \( a=-10 \), \( b=-3 \). So \(-3-(-10)+1 = 8 \). Let's list them: \(-10, -9, -8, -7, -6, -5, -4, -3\) (8 elements). So \(-9\) is in \( A \). Wait, so my earlier thought was wrong. So \(-9\) is in \( A \)? Wait, \(-9\) is greater than or equal to \(-10\) (yes, \(-9\geq -10\)) and less than or equal to \(-3\) (yes, \(-9\leq -3\)). So \(-9\in A\) is True? Wait, but let's re - check the set \( A \) definition: "integers greater than or equal to \(-10\) and less than or equal to \(-3\)". So \(-10,-9,-8,-7,-6,-5,-4,-3\): 8 elements. So \(-9\) is in \( A \). So \(-9\in A\) is True? Wait, but earlier when I thought about \(-9\), I made a mistake. So correction: \(-9\) is in \( A \), so \(-9\in A\) is True. Wait, but let's re - calculate \( n(A) \): from \(-10\) to \(-3\) inclusive. The number of integers is \((-3)-(-10)+1=-3 + 10 + 1=8\). So the elements are \(-10,-9,-8,-7,-6,-5,-4,-3\). So \(-9\) is in \( A \). So for \(-9\in A\), it is True. Wait, but earlier I was confused. So that's a mistake. So let's re - do the part (b) for \(-9\in A\):

• For \(-9\in A\): Since \( A=\{-10,-9,-8,-7,-6,-5,-4,-3\} \), \(-9\) is in \( A \), so this is True.

Final Answers

Part (a)

\( n(A)=\boldsymbol{8} \), \( n(B)=\boldsymbol{6} \)

Part (b)

• \(-23\in B\): True
• \(-16\in A\): False
• \(28

otin B\): True

• \(-9\in A\): True

(For the table, mark True for \(-23\in B\), False for \(-16\in A\), True for \(28
otin B\), True for \(-9\in A\))