Relations Test 22

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Relations Test 22

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Question 1
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$$(x, y)$$ and $$(p, q)$$ are two ordered pairs. Find the values of $$p$$ and $$y$$, if $$(4y + 5, 3p - 1) = (25, p + 1)$$

A
$$p = 0, y = 5$$

B
$$p = 1, y = 5$$

C
$$p = 0, y = 1$$

D
$$p = 1, y = 1$$

Solution

Given $$(x,y)=(p,q)$$
$$(4y + 5, 3p - 1) = (25, p + 1)$$
On equating we get
$$4y + 5 = 25$$
$$4y = 25 - 5$$
$$4y = 20$$
$$y = 5$$
$$3p - 1 = p + 1$$
$$3p - p = 1 + 1$$
$$2p = 2$$
$$p = 1$$
So, the value of$$ p = 1, y = 5$$
Question 2
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$$(x, y)$$ and $$(p, q)$$ are two ordered pairs. Find the values of $$x$$ and $$p$$, if $$(3x - 1, 9) = (11, p + 2)$$

A
$$x = 4, p = 9$$

B
$$x = 6, p = 7$$

C
$$x = 4, p = 5$$

D
$$x = 4, p = 7$$

Solution

Given$$(x,y)=(p,q)$$
$$(3x - 1, 9) = (11, p + 2)$$
By equating
$$3x - 1 = 11$$
$$3x = 12$$
$$x = 4$$
$$9 = p + 2$$
$$p = 7$$
So, the value of $$x = 4, p = 7.$$
Question 3
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If $$A \times B = \{(3, a), (3, -1), (3, 0), (5, a), (5, -1), (5, 0)\}$$, find $$A$$.

A
$$\{a, 5\}$$

B
$$\{a, -1\}$$

C
$$\{0, 5\}$$

D
$$\{3, 5\}$$

Solution
Question 4
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If $$A = \{2, 3\}$$ and $$B = \{1, 2\}$$, find $$A \times B$$.

A
$$\{(2, 1), (2, 2), (3, 1), (3, 2)\}$$

B
$$\{(2, 1), (2, 1), (3, 1), (3, 2)\}$$

C
$$\{(2, 1), (2, 2), (2, 1), (3, 2)\}$$

D
$$\{2, 1), (2, 2), (3, 1), (2, 2)\}$$

Solution
Question 5
1 / -0

If $$A \times B =$$ $${(2, 4), (2, a), (2, 5), (1, 4), (1, a), (1, 5)}$$, find $$B$$.

A
$$\{4, 2, 5\}$$

B
$$\{4, a, 5\}$$

C
$$\{4, 1, 5\}$$

D
$$\{2, a, 5\}$$

Solution
Question 6
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Let $$A = \left \{1, 2, 3\right \}$$. Then number of relations containing $$(1, 2)$$ and $$(1, 3)$$ which are reflexive and symmetric but not transitive is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution

Total possible pairs$$=\{ (1,1),(1,2),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\} $$
Reflexive means $$(a,a)$$ should be in relation.So $$(1,1),(2,2),(3,3)$$ should be in a relation.
Symmetric means if $$(a,b)$$ is in relation,then $$(b,a)$$ should be in relation.
So,since $$(1,2)$$ is in relation,$$(2,1)$$ should be in relation and since $$(1,3)$$ should be in relation.
Transitive means if $$(a,b)$$ is in relation,and $$(b,c)$$ is in relation,then $$(a,c)$$ is in relation.
If we add $$(2,3)$$,we need to add $$(3,2)$$ for symmetric,but it could become transitive then
Relation $$R1=\{ (1,1),(1,2),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\} $$
So,there is only $$1$$ possible relation.
Question 7
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What is the relation for the following diagram?

A
$$R =\{(2, 5), (2, 6), (3, 7)\}$$

B
$$R = \{(1, 5), (2, 6), (3, 7)\}$$

C
$$R =\{(1, 5), (2, 6), (1, 7)\}$$

D
$$R = \{(1, 5), (2, 6), (2, 7)\}$$

Solution

$$R:A\to B$$
$$A\times B = \{(1,5),(2,6),(3,7)\}$$
Question 8
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Which of the following do(es) not belong to $$A \times B$$ for the sets $$A = \{1, 2\}$$ and $$B =\{0, 2\}$$?

A
$$R = \{(1, 0), (2, 2)\}$$

B
$$R = \{(1, 1), (2, 1)\}$$

C
$$R = \{(1, 0), (1, 2)\}$$

D
$$R = \{(1, 2), (2, 2)\}$$

Solution

A relation for the sets $$A$$ and $$B$$ from $$A$$ to $$B$$ will be a subset of the $$A\times B.$$
Now the ordered pair belonging to $$A\times B$$ are
$$\{1,2\}\times\{0,2\}$$ $$=\{(1,0),(1,2),(2,0),(2,2)\}$$
Hence, $$\{(1,1)\}$$ and $$\{(2,1)\}$$ do not belong to $$A\times B$$.
Question 9
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Determine whether each of the following relations are reflexive, symmetric and transitive.

Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y): y is divisible by x}

A
R is reflexive

B
R is symmetric

C
R is transitive

D
None of these

Solution
Question 10
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Let $$S=\{(a, b, c) \in N \times N\times N:a+b+c=21, a \le b \le c\}$$ and $$T=\{(a, b, c)\in N\times N\times N: a, b, c\, are\, in\,AP\}$$, where $$N$$ is the set of all natural numbers. Then, the number of elements in the set $$S\cap T$$ is

A
$$6$$

B
$$7$$

C
$$13$$

D
$$14$$

Solution

We have, $$a+b+c=21$$ and $$\dfrac{a+c}{2}=b$$
$$\Rightarrow a+c+\dfrac{a+c}{2}=21$$
$$\Rightarrow a+c=14\Rightarrow \dfrac{a+c}{2}=7$$
$$\Rightarrow b=7$$
So, a can take values from $$1$$ to $$6$$ and $$c$$ can have values from $$8$$ to $$13$$
$$a=b=c=7$$
$$[\because a\le b \le c]$$
So, there are $$7$$ such triplets.