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The solution set of \( \Large 8x \equiv 6 \left(mod\ 14\right),\ x \epsilon z \) are:

A) \( \Large \left[ 8 \right] \cup \left[ 6 \right] \)
B) \( \Large \left[ 8 \right] \cup \left[ 14 \right] \)
C) \( \Large \left[ 6 \right] \cup \left[ 13 \right] \)
D) \( \Large \left[ 8 \right] \cup \left[ 6 \right] \cup \left[ 13 \

Correct Answer:



C) \( \Large \left[ 6 \right] \cup \left[ 13 \right] \)

Description for Correct answer:
Given that \( \Large 8x \equiv 6 \left(mod\ 14\right) \)

=> \( \Large 8x-6 = 14 P,\ \left(x\ \epsilon\ z\right) \)

=> \( \Large x = \frac{1}{8} \left(14P+6\right),\ \left(x\ \epsilon\ z\right) \)

=> \( \Large x = \frac{1}{4} \left(7P+3\right) \)

=> \( \Large x = 6,\ 13,\ 20,\ 27, 34,\ 41,\ 48... \)

Solution Set = = \( \Large x:x\ is\ a\ multiple\ of\ 12 \)

\( \Large \{ 6,\ 20,\ 34,\ 48... \} \cup\ \{ 13,\ 27,\ 41 \} \)

\( \Large \{ 6 \} \cup\ \{ 13 \} \)

Where \( \Large \left[ 6 \right],\ \left[ 13 \right] \) are equivalence classes of 6 and d13 respectively.

Part of solved Set theory questions and answers : >> Elementary Mathematics >> Set theory

Comments

Similar Questions

1). Consider the following relations:
\( \Large 1.\ A-B=A- \left(A \cap B\right) \)
\( \Large 2.\ A= \left(A \cap B\right) \cup \left(A-B\right) \)
\( \Large 3.\ A- \left(B \cup C\right) = \left(A-B\right) \cup \left(A-C\right) \)
which of these is/are correct?

A). (1) and (3)

B). (2) only

C). (2) and (3)

D). (1) and (2)

2). Let \( \Large f: \left(-1,\ 1\right)\rightarrow B \), be a function defined by \( \Large f \left(x\right)=\tan -1\frac{2x}{1-x^{2}} \), then is both one-one and onto when B is the interval:

A). \( \Large \left(-\frac{ \pi }{2},\ \frac{ \pi }{2}\right) \)

B). \( \Large \left[ -\frac{ \pi }{2},\ \frac{ \pi }{2} \right] \)

C). \( \Large \left[ 0,\ \frac{ \pi }{2} \right] \)

D). \( \Large \left(0,\ \frac{ \pi }{2}\right) \)

3). The function of \( \Large f \left(x\right)= log \left(x+\sqrt{x^{2}+1}\right) \) is:

A). an even function

B). an odd function

C). a periodic function

D). neither an even nor an odd function

4). If \( \Large f:R \rightarrow R \) satisfies \( \Large f \left(x+y\right) = f \left(x\right) = f \left(x\right)+f \left(y\right) \) for all \( \Large x,\ y,\ \epsilon R\ and\ f \left(1\right)=7 \), then \( \Large \sum^{n}_{r=1}f \left(r\right) \) is:

A). \( \Large \frac{7n}{2} \)

B). \( \Large \frac{7 \left(n+1\right) }{2} \)

C). \( \Large 7n \left(n+1\right) \)

D). \( \Large \frac{7n \left(n+1\right) }{2} \)

5). A function of from the set of natural numbers to integers defined by \( \Large f \left(n\right) = \frac{\frac{n-1}{2},\ when\ n\ is\ odd}{-\frac{n}{2},\ when\ n\ is\ even} \) is:

A). one-one but not onto

B). onto but not one-one

C). one-one and onto both

D). neither one-one nor onto

6). The relation "is subset of on the power set P(A) of set A is:"

A). symmetric

B). antisymmetric

C). equivalence relation

D). none of these

7). In a college of 300 students, every student read 5 newspaper and every newspaper is read by 60 students. The number of newspaper is:

A). at least 30

B). at most 20

C). exactly 25

D). none of these

8). Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basket ball, of the total 64 played both basket ball and hockey, 80 played cricket and basketball and 40 played cricket and hockey, 24 played all these three games. The numbers of boys who did not play any game is:

A). 128

B). 216

C). 240

D). 160

9). Let N denotes the set of all natural numbers and R be the relation on \( \Large N \times N \) defined by \( \Large \left(a,\ b\right) R \left(c,\ d\right)\ if\ ad \left(b+c\right)=bc \left(a+d\right) \), then R is:

A). symmetric only

B). reflexive only

C). transitive only

D). an equivalence relation

10). Let \( \Large f : \left[ 0,\ 1 \right] \rightarrow \left[ 0,\ 1 \right] \) defined by \( \Large f \left(x\right)=\frac{1-x}{1+x},\ 0 \le x \le 1 \) and let \( \Large g : \left[ 0,\ 1 \right] \rightarrow \left[ 0,\ 1 \right] \) be defined by \( \Large g \left(x\right) = 4x \left(1-x\right),\ 0 \le x \le 1 \), then fog and gof is:

A). \( \Large \frac{ \left(2x-1\right)^{2} }{1+4x+4x^{2}},\ \frac{8x \left(1-x\right) }{ \left(1+x\right)^{2} } \)

B). \( \Large \frac{ \left(2x-1\right) }{1+4x+4x^{2}},\ \frac{8 \left(1-x\right)x }{ \left(1+x\right)^{2} } \)

C). \( \Large \frac{ \left(2x+1\right)^{2} }{1+4x+4x^{2}},\ \frac{8}{ \left(1+x\right)^{2} } \)

D). \( \Large \frac{ \left(2x+1\right)^{2} }{ \left(1+x\right)^{2} },\ \frac{8 \left(1-x\right) }{ \left(1+x\right)^{2} } \)