Sets, Relations and Functions
In a committee $$50$$ people speak French, $$20$$ speak Spanish and $$10$$ speak both Spanish and French. The number of persons speaking at least one of these two languages is
If $$A = \{ 3, 5, 7, 9, 11 \}, B = \{7, 9, 11, 13\}, C = \{11, 13, 15\}$$ and $$D = \{15, 17\}; $$find (i) $$A \cap B$$ (ii) $$B \cap C$$ (iii) $$A \cap C \cap D$$ (iv) $$A \cap C$$ (v) $$B \cap D$$ (vi) $$A \cap (B \cup C)$$ (vii) $$A \cap D$$ (viii) $$A \cap (B \cup D)$$ (ix) $$( A \cap B ) \cap ( B \cap C )$$ (x) $$(A \cup D) \cap ( B \cup C)$$
$$A = \{ 3, 5, 7, 9, 11 \}, B = \{7, 9, 11, 13\}, C = \{11, 13, 15 \}$$ and $$D = \{15, 17\}$$
(i) $$A\cap B=\left \{7,9,11\right \}$$
(ii) $$B\cap C=\left \{11,13\right \}$$
(iii) $$A\cap C\cap D=(A\cap C ) \cap D=\left \{11\right \}\cap \left \{15, 17\right \}=\phi$$
(iv) $$A\cap C=\left \{11\right \}$$
(v) $$B\cap D=\phi$$
(vi) $$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$$
$$=\left \{7,9,11\right \}\cup \left \{11\right \}=\left \{7,9,11\right \}$$
(vii) $$A\cap D=\phi$$
(viii) $$A\cap (B\cup D)=(A\cap B)\cup (A\cap D)$$
$$=\left \{7,9,11\right \}\cup \phi =\left \{7,9,11\right \}$$
(ix) $$(A\cap B)\cap (B\cup C)=\left \{7,9,11\right \}\cap \left \{7,9,11,13,15\right \}=\left \{7,9,11\right \}$$
(x) $$(A\cup D)\cap (B\cup C)=\left \{3,5,7,9,11,15,17\right \}\cap \left \{7,9,11,13,15\right \}$$
$$=\left \{7,9,11,15\right \}$$
In a committee $$50$$ people speak French, $$20$$ speak Spanish and $$10$$ speak both Spanish and French. The number of persons speaking at least one of these two languages is
If $$A=\left\{ 1,2,3,4,5,6 \right\} $$, $$B=\left\{ 1,2 \right\} $$, then $$ \displaystyle \dfrac { A }{ \left( \dfrac { A }{ B }\right)} $$ is equal to
Let $$A=\left\{ 1,2,3,4 \right\} $$ and $$B=\left\{ 2,3,4,5,6 \right\} $$, then $$A\triangle B$$ is equal to =?$$$$ where, $$A\triangle B$$ is defined as $$\displaystyle {A}\triangle {B}=\{{x}:{x}\in\frac{{A}}{{B}}$$ or $$\displaystyle {x}\in\frac{{B}}{{A}}\})$$
If $$A, B$$ and $$C$$ are three sets such that $$A\cap B = A\cap C$$ and $$A\cup B = A\cup C$$, then
Let $$A=\{a, b\}, B = \{a, b, c\}.$$ Is $$A \subset B$$? What is $$A \cup B$$?
If A and B are two sets such that $$A \subset B$$, then what is $$A \cup B$$?
If $$A= \{1,2, 3, 4\}, B = \{3, 4, 5, 6\}, C = \{5, 6, 7, 8\}$$ and $$D = \{ 7, 8, 9, 10\}$$; find (i) $$A \cup B$$ (ii) $$A \cup C$$ (iii) $$B \cup C$$ (iv) $$B \cup D$$ (v) $$A \cup B \cup C$$ (vi) $$A \cup B \cup D$$ (vii) $$B \cup C \cup D$$
If $$X = \{ a, b, c, d \}$$ and $$Y = \{ f, b, d, g\}$$, find (i) $$X- Y$$ (ii) $$Y -X$$ (iii) $$X \cap Y$$
Let $$U = \left \{1, 2, 3, 4, 5, 6, 7, 8, 9 \right \}, A = \left \{ 1, 2, 3, 4\right \}, B = \left \{ 2, 4, 6, 8 \right \}$$ and $$C = \left \{ 3, 4, 5, 6 \right \}$$. Find (i) $$A'$$ (ii) $$B'$$ (iii) $$(A \cup C)'$$ (iv) $$(A \cup B)'$$ (v) $$(A')'$$ (vi) $$(B -C)'$$
Given that $$ \displaystyle \bar{X} $$ is the mean and $$ \displaystyle \sigma ^{2} $$ is the variance of $$ \displaystyle n $$ observations $$ \displaystyle X_{1},X_{2}...X_{n } $$. Prove that the mean and variance of the observations $$ \displaystyle aX_{1},aX_{2},aX_{3}....aX_{n } $$ are $$ \displaystyle \bar{ax} $$ and $$ \displaystyle a^{2}\sigma ^{2} $$ respectively $$ \displaystyle (a\neq 0) $$.