# Recent questions tagged sets

1 vote
1
For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ is defined as follows: $s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])$ Let $\mathcal{L}$ be a set of $10$-dimensional non-zero real vectors such that for every pair of distinct vectors $P,Q \in \mathcal{L}$, $s(P,Q)=0$. What is the maximum cardinality possible for the set $\mathcal{L}$? $9$ $10$ $11$ $100$
2
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A function $f$ from the set $A$ to itself is said to have a fixed point if $f(i)=i$ for some $i$ in $A$. Suppose $A$ is the set $\{a,b,c,d\}$. Find the number of bijective functions from the set $A$ to itself having no fixed point.
3
If $S$ be an infinite set and $S​_1​\dots\dots ,S​_n​$ be sets such that $S​_1 ​\cup S​_2​ \cup \dots \cup S​_n​ =S$, then at least one of the set $S​_i$​ is a finite set. not more than one of the sets $S​_i​$ can be finite. at least one of the sets $S​_i​$ is an infinite set. not more than one of the sets $S​_i​$ can be infinite.
4
Power set of empty set has exactly _______ subset One Two Zero Three
5
What is the Cardinality of the Power set of the set $\{0,1,2\}$? $8$ $6$ $7$ $9$
6
If $A$ and $B$ are two sets and $A \cup B = A ​ \cap ​ B$ then $A=\phi$ $B=\phi$ $A\neq B$ $A=B$
1 vote
7
X AND Y is an arbitrary sets, F: $X\rightarrow Y$ show that a and b are equivalent F is one-one For all set Z and function g1: $Z\rightarrow X$ and g2: $Z\rightarrow X$, if $g1 \neq g2$ implies $f \bigcirc g1 \neq f \bigcirc g2$ Where $\bigcirc$ is a fucntion composition.
8
If $A=\{x,y,z\}$ and $B=\{u,v,w,x\},$ and the universe is $\{s,t,u,v,w,x,y,z\}$. Then $(A \cup \overline{B}) \cap (A \cap B)$ is equal to $\{u,v,w,x\}$ $\{ \: \}$ $\{u,v,w,x,y,z\}$ $\{u,v,w\}$
1 vote
9
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1\} \:\:\:\:\:\:\: B=\{(x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is non-empty none of the above
10
Let $\mathbb{N}=\{1,2,3, \dots\}$ be the set of natural numbers. For each $n \in \mathbb{N}$, define $A_n=\{(n+1)k, \: k \in \mathbb{N} \}$. Then $A_1 \cap A_2$ equals $A_3$ $A_4$ $A_5$ $A_6$
1 vote
11
Let $A$ and $B$ be disjoint sets containing $m$ and $n$ elements respectively, and let $C=A \cup B$. Then the number of subsets $S$ (of $C$) which contains $p$ elements and also has the property that $S \cap A$ contains $q$ ... $\begin{pmatrix} m \\ p-q \end{pmatrix} \times \begin{pmatrix} n \\ q \end{pmatrix}$
1 vote
12
A set contains $2n+1$ elements. The number of subsets of the set which contain at most $n$ elements is $2^n$ $2^{n+1}$ $2^{n-1}$ $2^{2n}$
1 vote
13
Let $X$ be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. Define the set $\mathcal{R}$ by $\mathcal{R} = \{(x,y) \in X \times X : x$ and $y$ have the same remainder when divided by $3\}$. Then the number of elements in $\mathcal{R}$ is $40$ $36$ $34$ $33$
1 vote
Let $A$ be a set of $n$ elements. The number of ways, we can choose an ordered pair $(B,C)$, where $B,C$ are disjoint subsets of $A$, equals $n^2$ $n^3$ $2^n$ $3^n$
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ by $f(x,A)=\begin{cases} 1 & \text{ if } x \in A \\ 0 & \text{ if } x \notin A \end{cases}$ Then $f(x, A \cup B)$ equals $f(x,A)+f(x,B)$ $f(x,A)+f(x,B)\: – 1$ $f(x,A)+f(x,B)\: – f(x,A) \cdot f(x,B)$ $f(x,A)\:+ \mid f(x,A)\: – f(x,B) \mid$
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is non-empty none of the above