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GO Classes Set Theory And Algebra Practice Set 1 | Question: 3
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The following table describes a binary relation. Find the set of ordered pairs that is this relation, as in the definition of a binary relation.
$$
\begin{array}{c|c|c|c|c|c|c|}
\sim & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline 1 & \ast & & & & & \ast \\
\hline 2 & & \ast & & & & \\
\hline 3 & & & & \ast & \ast & \\
\hline 4 & & & \ast & & \ast & \\
\hline 5 & & & \ast & \ast & & \\
\hline 6 & \ast & & & & & \ast \\
\hline
\end{array}$$
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Ordered pairs of this relation :- (1,1),(1,6),(2,2),(3,4),(3,5),(4,3),(4,5),(5,3),(5,4),(6,1),(6,6).
total ordered pairs = 11
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The ordered pair of the given relation is as follows:
$R= \left\{ (1,1)\ (1,6)\ (2,2)\ (3,4)\ (3,5)\ (4,3)\ (4,5) \ (5,3)\ (5,4)\ (6,1)\ (6,6)\right \}$
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Jun 2, 2023
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GO Classes Set Theory And Algebra Practice Set 1 | Question: 1
Let $A=\{0,1,2,3\}$ and $R$ a relation over $A$ : $ R=\{(0,0),(0,1),(0,3),(1,1),(1,0),(2,3),(3,3)\} $ Draw the directed graph of $R$. Check whether $R$ is an equivalence relation. Give a counterexample in each case in which the relation does not satisfy one of the properties of being an equivalence relation.
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GO Classes Set Theory And Algebra Practice Set 1 | Question: 2
$\begin{array}{l|llllll} \textbf{Relations on}\; \mathbb{Z}: & \quad < & \qquad \leq & \qquad = & \qquad \mid & \qquad \nmid & \qquad \neq \\ \hline \hline \text{Reflexive} & \\ \text{Symmetric} & \\ \text{Transitive} & \end{array}$ Fill "Yes or No" in the above table.
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GO Classes Set Theory And Algebra Practice Set 1 | Question: 6
Let $A$ be any set. Subset Relation on $\mathrm{P}(\mathrm{A})$ is Anti-symmetric?
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GO Classes Set Theory And Algebra Practice Set 1 | Question: 7
Define $\mathcal{R}$ the binary relation on $\mathbb{N} \times \mathbb{N}$ to mean $(a, b) \mathcal{R}(c, d)$ iff $b \mid d$ and $a \mid c$
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