Let $$A, B$$ and $$C$$ be sets such that $$\phi = A\cap B \subseteq C$$. Then which of the following statements is not true?
for $$A = C, A - C = \phi$$
$$\Rightarrow \phi \subseteq B$$
But $$A\not {\subseteq} B$$
$$\Rightarrow$$ option $$1$$ is NOT true
Let $$x \epsilon (C x \epsilon (C \cup A)\cap (C\cup B)$$
$$\Rightarrow x\epsilon (C\cup A)$$ and $$x \epsilon (C\cup B)$$
$$\Rightarrow (x \epsilon C$$ or $$x \epsilon A)$$ and $$(x\epsilon C$$ or $$x \epsilon B)$$
$$\Rightarrow x \epsilon C$$ or $$x \epsilon (A\cap B)$$
$$\Rightarrow x \epsilon C$$ or $$x\epsilon C$$ (as $$A\cup B\subseteq C)$$
$$\Rightarrow x \epsilon C$$
$$\Rightarrow (C \cup A)\cap (C\cup B)\subseteq ....(1)$$
Now $$x \epsilon C\Rightarrow x \epsilon (C\cup A)$$ and $$x \epsilon (C \cup B)$$
$$\Rightarrow x\epsilon (C\cup A)\cap (C\cup B)$$
$$\Rightarrow C\subseteq (C\cup A)\cap (C \cup B) .....(2)$$
$$\Rightarrow$$ from (1) and (2)
$$C = (C\cup A)\cap (C\cup B)$$
$$\Rightarrow$$ option 2 is true
Let $$x \epsilon A$$ and $$x \not {\epsilon} B$$
$$\Rightarrow x \epsilon (A - B)$$
$$\Rightarrow x \epsilon C$$ (as $$A - B \subseteq C)$$
Let $$x \epsilon A$$ and $$x \epsilon B$$
$$\Rightarrow x \epsilon (A\cap B)$$
$$\Rightarrow x \epsilon C$$ (as $$A\cap B\subseteq C)$$
Hence $$x \epsilon A \Rightarrow x \epsilon C$$
$$\Rightarrow A \subseteq C$$
$$\Rightarrow$$ Option 3 is true
as $$C\supseteq (A\cap B)$$
$$\Rightarrow B\cap C\supseteq (A\cap B)$$
as $$A\cap B\neq \phi$$
$$\Rightarrow B\cap C \neq \phi$$
$$\Rightarrow$$ Option 4 is true.
State true or false. Given universal set= $$\displaystyle =\left \{ -6,-5\frac{3}{4}, -\sqrt{4}, -\frac{3}{5}, -\frac{3}{8}, 0, \frac{4}{5}, 1, 1\frac{2}{3}, \sqrt{8}, 3.01, \pi , 8.47 \right \}$$ From the given set, the set of integers is $$\displaystyle \left \{ -6,-\sqrt{4}, 0,1 \right \}$$.
State true or false. Given universal set= $$\displaystyle =\left \{ -6,-5\frac{3}{4}, -\sqrt{4}, -\frac{3}{5}, -\frac{3}{8}, 0, \frac{4}{5}, 1, 1\frac{2}{3}, \sqrt{8}, 3.01, \pi , 8.47 \right \}$$ From the given set, find set of non-negative integers is $$\displaystyle \left \{0,1 \right \}$$.
State true or false: A set of rational number is a subset of a set of real numbers.
What universal set (s) would you propose for each of the following: The set of isosceles triangles.
Given a non empty set X, consider $$P(X)$$ which is set of all subsets of $$X$$. Define the relation $$R$$ is $$P(X)$$ as follows: For subsets $$A, B$$ in $$P(X), ARB$$ if and only if $$A\subset B$$. Is R an equivalence relation on $$P(X)$$? Justify your answer
Let X be a set of $$5$$ elements. The number d of ordered pairs (A, B) of subsets of X such that $$A\neq \Phi, B\neq \Phi, A\cap B=\Phi$$ satisfies.
For any set $$A$$, if $$A\subseteq \phi \Leftrightarrow A=\phi$$.
Examine whether the following statements are true or false: $$(a,b)\not{\subset}(b,c,a)$$
Examine whether the following statements are true or false: $$(a,e) \subset$$ ($$x : x$$ is a vowel in the English alphabet)
Examine whether the following statements are true or false: $$(1,2,3)\subset (1,3,5)$$