Sure, let's find all possible binary relations for each of the given cases.
### Part (i)
Sets:
[tex]\( A = \{\sqrt{2}, \sqrt{3}, \sqrt{5}\} \)[/tex]
[tex]\( B = \{\sqrt[3]{5}\} \)[/tex]
Step-by-Step:
1. The number of elements in set [tex]\( A \)[/tex] is 3: [tex]\(\sqrt{2}\)[/tex], [tex]\(\sqrt{3}\)[/tex], [tex]\(\sqrt{5}\)[/tex].
2. The number of elements in set [tex]\( B \)[/tex] is 1: [tex]\(\sqrt[3]{5}\)[/tex].
3. The total number of binary relations from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is determined by the number of ordered pairs [tex]\((a,b)\)[/tex], where [tex]\( a \in A \)[/tex] and [tex]\( b \in B \)[/tex].
4. The number of possible ordered pairs is the product of the number of elements in [tex]\( A \)[/tex] and [tex]\( B \)[/tex], which is [tex]\( 3 \times 1 = 3 \)[/tex].
5. Each of these ordered pairs can either be included or not in a binary relation, giving us [tex]\(2\)[/tex] choices (include or exclude) for each of the 3 pairs.
6. Therefore, the total number of binary relations is [tex]\( 2^3 = 8 \)[/tex].
### Part (ii)
Set:
[tex]\( C = \{\pi, e\} \)[/tex]
Step-by-Step:
1. The number of elements in set [tex]\( C \)[/tex] is 2: [tex]\(\pi\)[/tex], [tex]\(e\)[/tex].
2. Here, we are considering binary relations from [tex]\( C \)[/tex] to [tex]\( C \)[/tex].
3. The number of possible ordered pairs [tex]\((c_1, c_2)\)[/tex], where [tex]\( c_1, c_2 \in C \)[/tex], is [tex]\( 2 \times 2 = 4 \)[/tex].
4. Each of these ordered pairs can either be included or not in a binary relation, giving us [tex]\(2\)[/tex] choices (include or exclude) for each of the 4 pairs.
5. Therefore, the total number of binary relations is [tex]\( 2^4 = 16 \)[/tex].
### Part (iii)
Sets:
[tex]\( D = \{5\} \)[/tex]
[tex]\( E = \{1, 10\} \)[/tex]
Step-by-Step:
1. The number of elements in set [tex]\( D \)[/tex] is 1: [tex]\(5\)[/tex].
2. The number of elements in set [tex]\( E \)[/tex] is 2: [tex]\(1\)[/tex], [tex]\(10\)[/tex].
3. The total number of binary relations from [tex]\( D \)[/tex] to [tex]\( E \)[/tex] is determined by the number of ordered pairs [tex]\((d,e)\)[/tex], where [tex]\( d \in D \)[/tex] and [tex]\( e \in E \)[/tex].
4. The number of possible ordered pairs is the product of the number of elements in [tex]\( D \)[/tex] and [tex]\( E \)[/tex], which is [tex]\( 1 \times 2 = 2 \)[/tex].
5. Each of these ordered pairs can either be included or not in a binary relation, giving us [tex]\(2\)[/tex] choices (include or exclude) for each of the 2 pairs.
6. Therefore, the total number of binary relations is [tex]\( 2^2 = 4 \)[/tex].
### Conclusion
(i) The number of binary relations from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is [tex]\( 8 \)[/tex].
(ii) The number of binary relations from [tex]\( C \)[/tex] to [tex]\( C \)[/tex] is [tex]\( 16 \)[/tex].
(iii) The number of binary relations from [tex]\( D \)[/tex] to [tex]\( E \)[/tex] is [tex]\( 4 \)[/tex].
