11. the symbol $ \subseteq $ means:
a. proper subset
b. subset
c. union
d. intersection

12. in a venn diagram, $ a \cap b $ represents:
a. only in a
b. only in b
c. in both a and b
d. outside both

13. which of the following is irrational?
a. $ \frac{2}{3} $
b. $ \sqrt{9} $
c. $ \sqrt{7} $
d. 0.5

14. the additive inverse of $ -5 $ is:
a. 5
b. $ -5 $
c. $ \frac{1}{5} $
d. $ -\frac{1}{5} $

15. if $ u = \{1,2,3,4,5\} $ and $ a = \{1,3,5\} $, find $ a $.
a. $ \{1,3,5\} $
b. $ \{2,4\} $
c. $ u $
d. $ \emptyset $

16. which shows associative property of multiplication?
a. $ 3 \times 4 = 4 \times 3 $
b. $ (2 \times 3) \times 4 = 2 \times (3 \times 4) $
c. $ 2(3 + 4) = 6 + 8 $
d. $ 5 \times 1 = 5 $

17. simplify $ 6(5 + 2) $.
a. 32
b. 42
c. 30
d. 12

18. if $ a = \{a,b,c\} $ and $ b = \{b,c,d,e\} $, find $ a \cup b $.
a. $ \{a,b,c,d,e\} $
b. $ \{b,c\} $
c. $ \{a,d,e\} $
d. $ \emptyset $

19. rational numbers are:
a. closed under division
b. closed under subtraction
c. all irrational
d. not fractions

20. $ n(a \cap b \cap c) $ represents:
a. in a only
b. in all three sets
c. in a and b only
d. outside all

21. identify the property: $ 7 + 0 = 7 $.
a. commutative
b. associative
c. identity
d. inverse

22. if $ x = \{2,4,6,8\} $ and $ y = \{4,8,12\} $:
a. $ x \subset y $
b. $ y \subset x $
c. disjoint
d. $ x \cap y = \{4,8\} $

23. using distributive property, $ 9 \times 11 = $
a. 91
b. 99
c. 110
d. 20

24. proper subset of $ \{1,2,3,4\} $ is:
a. $ \{1,2,3,4\} $
b. $ \{1,2\} $
c. $ \{5,6\} $
d. $ \{1,2,3,4,5\} $

25. if $ a = \{2,4,6\} $, then $ a \cup a = $
a. a
b. $ a $
c. u
d. $ \emptyset $

28. $ p \cap q $ where p, q are factors of 12 and 18:
a. $ \{1,2,3,6\} $
b. $ \{12,18\} $
c. $ \{4,9\} $
d. $ \{1,2,3,4,6,9\} $

29. integers are not closed under division because:
a. always whole
b. may give fractions
c. division is commutative

Question

11. the symbol $ \subseteq $ means:
a. proper subset
b. subset
c. union
d. intersection

12. in a venn diagram, $ a \cap b $ represents:
a. only in a
b. only in b
c. in both a and b
d. outside both

13. which of the following is irrational?
a. $ \frac{2}{3} $
b. $ \sqrt{9} $
c. $ \sqrt{7} $
d. 0.5

14. the additive inverse of $ -5 $ is:
a. 5
b. $ -5 $
c. $ \frac{1}{5} $
d. $ -\frac{1}{5} $

15. if $ u = \{1,2,3,4,5\} $ and $ a = \{1,3,5\} $, find $ a $.
a. $ \{1,3,5\} $
b. $ \{2,4\} $
c. $ u $
d. $ \emptyset $

16. which shows associative property of multiplication?
a. $ 3 \times 4 = 4 \times 3 $
b. $ (2 \times 3) \times 4 = 2 \times (3 \times 4) $
c. $ 2(3 + 4) = 6 + 8 $
d. $ 5 \times 1 = 5 $

17. simplify $ 6(5 + 2) $.
a. 32
b. 42
c. 30
d. 12

18. if $ a = \{a,b,c\} $ and $ b = \{b,c,d,e\} $, find $ a \cup b $.
a. $ \{a,b,c,d,e\} $
b. $ \{b,c\} $
c. $ \{a,d,e\} $
d. $ \emptyset $

19. rational numbers are:
a. closed under division
b. closed under subtraction
c. all irrational
d. not fractions

20. $ n(a \cap b \cap c) $ represents:
a. in a only
b. in all three sets
c. in a and b only
d. outside all

21. identify the property: $ 7 + 0 = 7 $.
a. commutative
b. associative
c. identity
d. inverse

22. if $ x = \{2,4,6,8\} $ and $ y = \{4,8,12\} $:
a. $ x \subset y $
b. $ y \subset x $
c. disjoint
d. $ x \cap y = \{4,8\} $

23. using distributive property, $ 9 \times 11 = $
a. 91
b. 99
c. 110
d. 20

24. proper subset of $ \{1,2,3,4\} $ is:
a. $ \{1,2,3,4\} $
b. $ \{1,2\} $
c. $ \{5,6\} $
d. $ \{1,2,3,4,5\} $

25. if $ a = \{2,4,6\} $, then $ a \cup a = $
a. a
b. $ a $
c. u
d. $ \emptyset $

28. $ p \cap q $ where p, q are factors of 12 and 18:
a. $ \{1,2,3,6\} $
b. $ \{12,18\} $
c. $ \{4,9\} $
d. $ \{1,2,3,4,6,9\} $

29. integers are not closed under division because:
a. always whole
b. may give fractions
c. division is commutative

Accepted Answer

### Question 11
# Brief Explanations:
The symbol $\subseteq$ is defined as "subset", meaning one set is a subset (can be equal or proper) of another. A proper subset is $\subset$, union is $\cup$, intersection is $\cap$.
# Answer:
B. Subset

### Question 12
# Brief Explanations:
In a Venn diagram, $A\cap B$ (intersection) represents the elements that are in both set $A$ and set $B$. "Only in $A$" is $A - B$, "Only in $B$" is $B - A$, "Outside both" is the complement of $A\cup B$.
# Answer:
C. In both A and B

### Question 13
# Brief Explanations:
Irrational numbers are non - repeating, non - terminating decimals and cannot be expressed as a fraction of two integers. $\frac{2}{3}$ is a fraction (rational), $\sqrt{9}=3$ (rational), $0.5=\frac{1}{2}$ (rational), and $\sqrt{7}$ is non - rational (irrational as 7 is not a perfect square).
# Answer:
C. $\sqrt{7}$

### Question 14
# Brief Explanations:
The additive inverse of a number $x$ is a number $y$ such that $x + y=0$. For $x = - 5$, $y = 5$ since $-5 + 5=0$.
# Answer:
A. 5

### Question 15
# Brief Explanations:
The complement of a set $A$ ($A'$) with respect to the universal set $U$ is the set of all elements in $U$ that are not in $A$. Given $U=\{1,2,3,4,5\}$ and $A = \{1,3,5\}$, the elements in $U$ but not in $A$ are $\{2,4\}$.
# Answer:
B. $\{2,4\}$

### Question 16
# Brief Explanations:
The associative property of multiplication states that $(a	imes b)	imes c=a	imes(b	imes c)$. Option A is commutative property ($a	imes b = b	imes a$), option C is an incorrect application (should be $2(3 + 4)=6+8$ is wrong, correct is $2(3 + 4)=6 + 8$? No, $2(3 + 4)=2	imes3+2	imes4 = 6 + 8$? Wait, no, $2(3 + 4)=2	imes7 = 14$ and $6 + 8 = 14$? But the form is distributive. Wait, the associative property for multiplication is $(2	imes3)	imes4=2	imes(3	imes4)$, which is option B.
# Answer:
B. $(2	imes3)	imes4 = 2	imes(3	imes4)$

### Question 17
# Explanation:
## Step 1: Apply the order of operations (PEMDAS) - first parentheses
We first simplify the expression inside the parentheses: $5 + 2=7$.
## Step 2: Multiply
Then we multiply 6 by the result: $6	imes7 = 42$.
# Answer:
B. 42

### Question 18
# Brief Explanations:
The union of two sets $A\cup B$ is the set of all elements that are in $A$ or in $B$ (or in both). Given $A=\{a,b,c\}$ and $B = \{b,c,d,e\}$, combining the elements (without repetition) gives $\{a,b,c,d,e\}$.
# Answer:
A. $\{a,b,c,d,e\}$

### Question 19
# Brief Explanations:
Rational numbers are closed under subtraction (if $a$ and $b$ are rational, $a - b$ is rational). They are not closed under division (e.g., $1\div0$ is undefined, and $2\div4=\frac{1}{2}$ is rational but $1\div2=\frac{1}{2}$ is rational, but $0\div0$ is undefined. Wait, more accurately, non - zero rational numbers are closed under division? No, the set of rational numbers: if we consider division, when the divisor is 0, it's not defined. But for subtraction, for any two rational numbers $\frac{p}{q}$ and $\frac{r}{s}$ ($q
eq0,s
eq0$), $\frac{p}{q}-\frac{r}{s}=\frac{ps - rq}{qs}$, which is a rational number (since $ps - rq$ and $qs$ are integers and $qs
eq0$). Also, rational numbers are not all irrational (they are the opposite), and they can be fractions. So rational numbers are closed under subtraction.
# Answer:
B. Closed under subtraction

### Question 20
# Brief Explanations:
$n(A\cap B\cap C)$ represents the number of elements that are in set $A$, set $B$, and set $C$ (in all three sets). "In $A$ only" is $n(A-(B\cup C))$, "In $A$ and $B$ only" is $n((A\cap B)-C)$, "Outside all" is $n((A\cup B\cup C)')$.
# Answer:
B. In all three sets

### Question 21
# Brief Explanations:
The identity property of addition states that adding 0 to a number leaves the number unchanged, i.e., $a + 0=a$. The commutative property is $a + b=b + a$, associative is $(a + b)+c=a+(b + c)$, inverse is $a+(-a)=0$. Here $7 + 0 = 7$ follows the identity property.
# Answer:
C. Identity

### Question 22
# Brief Explanations:
To find the intersection $X\cap Y$, we look for elements common to both $X=\{2,4,6,8\}$ and $Y = \{4,8,12\}$. The common elements are 4 and 8, so $X\cap Y=\{4,8\}$. $X$ is not a subset of $Y$ (2,6 are in $X$ not in $Y$), $Y$ is not a subset of $X$ (12 is in $Y$ not in $X$), and they are not disjoint (they have common elements).
# Answer:
D. $X\cap Y=\{4,8\}$

### Question 23
# Explanation:
## Step 1: Apply the distributive property
The distributive property is $a	imes(b + c)=a	imes b+a	imes c$. We can write $11$ as $10 + 1$, so $9	imes11=9	imes(10 + 1)$.
## Step 2: Multiply and add
Using the distributive property: $9	imes(10 + 1)=9	imes10+9	imes1=90 + 9 = 99$.
# Answer:
B. 99

### Question 24
# Brief Explanations:
A proper subset of a set $S$ is a subset that is not equal to $S$. For the set $\{1,2,3,4\}$, $\{1,2\}$ is a proper subset (it is a subset and not equal to the original set). $\{1,2,3,4\}$ is the set itself (not a proper subset), $\{5,6\}$ is not a subset (elements not in the original set), $\{1,2,3,4,5\}$ is not a subset (5 is not in the original set).
# Answer:
B. $\{1,2\}$

### Question 25
# Brief Explanations:
For any set $A$, the union of $A$ and its complement $A'$ is the universal set $U$. If $A=\{2,4,6\}$, $A'$ is the set of all elements not in $A$ (with respect to the universal set), and $A\cup A'=U$.
# Answer:
C. $U$

### Question 28
# Explanation:
## Step 1: Find factors of 12 and 18
The factors of 12 are $1,2,3,4,6,12$. The factors of 18 are $1,2,3,6,9,18$.
## Step 2: Find the intersection
The intersection $P\cap Q$ (where $P$ is factors of 12 and $Q$ is factors of 18) is the set of common factors, which is $\{1,2,3,6\}$.
# Answer:
A. $\{1,2,3,6\}$

### Question 29
# Brief Explanations:
Integers are not closed under division because when we divide two integers, we may get a non - integer (a fraction). For example, $2\div4=\frac{1}{2}$ (a fraction). Division is not commutative (e.g., $4\div2
eq2\div4$), and the fact that division may give whole numbers sometimes does not mean it is closed (the closure property requires that the result is always in the set). So integers are not closed under division because they may give fractions.
# Answer:
B. May give fractions

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