# Binary Relations

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

The binary relation "less than or equal to" is defined on the set A = {0, 2, 5, 7}. Represent it in roster form.

### Example 2

The binary relation "greater than" is defined on the set $$B = \left\{ {\frac{2}{3}, \frac{4}{7}, \frac{5}{8}, \frac{{11}}{{17}}} \right\}.$$ Represent it in roster form.

### Example 3

Let $$A = \left\{ {1,3,4,5,7} \right\}.$$ Represent the relation $$R = \left\{ {\left( {a,b} \right) \mid \left| {a - b} \right| \lt 2} \right\}$$ on the set $$A$$ in matrix form.

### Example 4

Let $$A = \left\{ {1,2,3,4,5,6} \right\}.$$ Represent the divisibility relation on the set $$A$$ as a matrix.

### Example 5

The relation $$R$$ on the set $$C = \left\{ {0,1,2,3} \right\}$$ is given by the matrix $M = \left[ {\begin{array}{*{20}{c}} 1&0&1&1\\ 0&1&0&1\\ 0&1&0&0\\ 0&1&0&1 \end{array}} \right].$ Represent $$R$$ using a digraph.

### Example 6

Let $$B = \left\{ {2,4,5,7,8,9} \right\}.$$ A relation $$R$$ on the set $$B$$ is defined as follows: $R = \left\{ {\left( {a,b} \right) \mid 2 \mid \left( {a + b} \right)} \right\}.$ Draw the digraph of $$R.$$

### Example 1.

The binary relation "less than or equal to" is defined on the set $$A = \left\{ {0,2,5,7} \right\}.$$ Represent it in roster form.

Solution.

We list all ordered pairs that satisfy the relation definition:

$R = \left\{ {\left( {0,0} \right),\left( {0,2} \right),\left( {0,5} \right),\left( {0,7} \right),\left( {2,2} \right),\left( {2,5} \right),\left( {2,7} \right),\left( {5,5} \right),\left( {5,7} \right),\left( {7,7} \right)} \right\}.$

### Example 2.

The binary relation "greater than" is defined on the set $$B = \left\{ {\frac{2}{3}, \frac{4}{7}, \frac{5}{8}, \frac{{11}}{{17}}} \right\}.$$ Represent it in roster form.

Solution.

Given that

$\frac{2}{3} \approx 0.666,\;\;\frac{4}{7} \approx 0.571,\;\;\frac{5}{8} = 0.625,\;\;\frac{{11}}{{17}} \approx 0.647,$

we list the elements of the set $$B$$ in increasing order:

$B = \left\{ {\frac{4}{7},\frac{5}{8},\frac{{11}}{{17}},\frac{2}{3} }\right\}.$

Now we can build the ordered pairs satisfying the relation "greater than":

$R = \left\{ {\left( {\frac{5}{8},\frac{4}{7}} \right),\left( {\frac{{11}}{{17}},\frac{4}{7}} \right),\left( {\frac{2}{3},\frac{4}{7}} \right),\left( {\frac{{11}}{{17}},\frac{5}{8}} \right),\left( {\frac{2}{3},\frac{5}{8}} \right),\left( {\frac{2}{3},\frac{{11}}{{17}}} \right)} \right\}.$

### Example 3.

Let $$A = \left\{ {1,3,4,5,7} \right\}.$$ Represent the relation $$R = \left\{ {\left( {a,b} \right) \mid \left| {a - b} \right| \lt 2} \right\}$$ on the set $$A$$ in matrix form.

Solution.

The logical matrix $$M$$ represents all ordered pairs $${\left( {a,b} \right)}$$ on the set $$A.$$ If an element of the matrix (an ordered pair) satisfies the condition $${\left| {a - b} \right| \lt 2},$$ it is labeled by $$1.$$ If not, it is labeled by $$0.$$

### Example 4.

Let $$A = \left\{ {1,2,3,4,5,6} \right\}.$$ Represent the divisibility relation on the set $$A$$ as a matrix.

Solution.

The $$\left({i,j}\right)-$$entry of the logical matrix is set to be equal to $$1,$$ if the element of the $$i\text{th}$$ row divides the element of the $$j\text{th}$$ column. Otherwise, the $$\left({i,j}\right)-$$entry is zero.

### Example 5.

The relation $$R$$ on the set $$C = \left\{ {0,1,2,3} \right\}$$ is given by the matrix $M = \left[ {\begin{array}{*{20}{c}} 1&0&1&1\\ 0&1&0&1\\ 0&1&0&0\\ 0&1&0&1 \end{array}} \right].$ Represent $$R$$ using a digraph.

Solution.

The digraph contains $$4$$ nodes and $$8$$ directed edges. The number of nodes is equal to the cardinality of the set $$C,$$ and the number of edges is equal to the number of matrix elements with value $$1.$$

### Example 6.

Let $$B = \left\{ {2,4,5,7,8,9} \right\}.$$ A relation $$R$$ on the set $$B$$ is defined as follows: $R = \left\{ {\left( {a,b} \right) \mid 2 \mid \left( {a + b} \right)} \right\}.$ Draw the digraph of $$R.$$

Solution.

To satisfy the divisibility condition $${2 \mid \left( {a + b} \right)},$$ the sum of the components $$a$$ and $$b$$ must be even. Hence, the relation $$R$$ includes the following ordered pairs:

$R = \left\{ {\left( {2,2} \right),\left( {2,4} \right),\left( {2,8} \right),\left( {4,2} \right),\left( {4,4} \right),\left( {4,8} \right),\left( {5,5} \right),\left( {5,7} \right),\left( {5,9} \right),\left( {7,5} \right),\left( {7,7} \right),\left( {7,9} \right),\left( {8,2} \right),\left( {8,4} \right),\left( {8,8} \right),\left( {9,5} \right),\left( {9,7} \right),\left( {9,9} \right)} \right\}.$

The directed graph of the divisibility relation looks as follows: