Answer :
To solve the problem, we need to fill in the missing values such that each pair in the first relation corresponds to an inverted pair in the second relation. Below is the step-by-step solution:
1. Identify and fill in the missing values for [tex]\( a \)[/tex]:
- In the first relation, the pair at the second position is [tex]\((None, 3.2)\)[/tex].
- In the second relation, the pair corresponding to [tex]\((x, y)\)[/tex] for [tex]\( x = -3.8 \)[/tex] from the first relation should be [tex]\((y, x)\)[/tex].
- Therefore, the value corresponding to [tex]\( x = -3.8 \)[/tex] in the first relation should match [tex]\( y = 3.2 \)[/tex] in the second relation, confirming that [tex]\( a = 3.2 \)[/tex].
2. Identify and fill in the missing values for [tex]\( b \)[/tex]:
- In the first relation, the pair at the first position is [tex]\((-3.8, -3.1)\)[/tex].
- In the second relation, the pair with [tex]\( x = -3.1 \)[/tex] should have its [tex]\( y \)[/tex] value as [tex]\( -3.8 \)[/tex].
- Therefore, for [tex]\( x = -3.1 \)[/tex], we need [tex]\( y = -3.1 \)[/tex] in the first relation and [tex]\( y = -3.1 \)[/tex] in the second relation, confirming that [tex]\( b = -3.1 \)[/tex].
3. Identify and fill in the missing values for [tex]\( c \)[/tex]:
- In the first relation, the pair at the third position is [tex]\((-1.4, None)\)[/tex].
- In the second relation, the pair with [tex]\( x = 1.7 \)[/tex] and a counterpart [tex]\( y = -1.4 \)[/tex].
- Therefore, the value corresponding to [tex]\( x = 1.7 \)[/tex] in the second relation should match [tex]\( y = -1.4 \)[/tex] in the first relation, confirming that [tex]\( c = 3.2 \)[/tex].
4. Identify and fill in the missing values for [tex]\( d \)[/tex]:
- In the first relation, the pair at the fourth position is [tex]\((-0.2, 4.4)\)[/tex].
- In the second relation, the [tex]\( y \)[/tex] for [tex]\((-0.2)\)[/tex] is not known yet.
- Therefore, the position corresponding in the first relation should give us the value directly: since there's no existing pair just in the entered set information, we can't deduce [tex]\( d \)[/tex]. So, [tex]\( d = None \)[/tex].
5. Identify and fill in the missing values for [tex]\( e \)[/tex]:
- In the first relation, the pair at the fifth position is [tex]\((1.0, 5.0)\)[/tex].
- In the second relation, the pair with [tex]\( x = 5.0 \)[/tex] should have [tex]\( y \)[/tex] matching [tex]\( x's \)[/tex] partner in the first relation, confirming that [tex]\( e = 5.0 \)[/tex].
So, the values that make the two relations inverses of each other are:
- [tex]\( a = 3.2 \)[/tex]
- [tex]\( b = -3.1 \)[/tex]
- [tex]\( c = 3.2 \)[/tex]
- [tex]\( d = None \)[/tex]
- [tex]\( e = 5.0 \)[/tex]
Thus:
[tex]\[ \boxed{(a, b, c, d, e) = (3.2, -3.1, 3.2, \text{None}, 5.0)} \][/tex]
1. Identify and fill in the missing values for [tex]\( a \)[/tex]:
- In the first relation, the pair at the second position is [tex]\((None, 3.2)\)[/tex].
- In the second relation, the pair corresponding to [tex]\((x, y)\)[/tex] for [tex]\( x = -3.8 \)[/tex] from the first relation should be [tex]\((y, x)\)[/tex].
- Therefore, the value corresponding to [tex]\( x = -3.8 \)[/tex] in the first relation should match [tex]\( y = 3.2 \)[/tex] in the second relation, confirming that [tex]\( a = 3.2 \)[/tex].
2. Identify and fill in the missing values for [tex]\( b \)[/tex]:
- In the first relation, the pair at the first position is [tex]\((-3.8, -3.1)\)[/tex].
- In the second relation, the pair with [tex]\( x = -3.1 \)[/tex] should have its [tex]\( y \)[/tex] value as [tex]\( -3.8 \)[/tex].
- Therefore, for [tex]\( x = -3.1 \)[/tex], we need [tex]\( y = -3.1 \)[/tex] in the first relation and [tex]\( y = -3.1 \)[/tex] in the second relation, confirming that [tex]\( b = -3.1 \)[/tex].
3. Identify and fill in the missing values for [tex]\( c \)[/tex]:
- In the first relation, the pair at the third position is [tex]\((-1.4, None)\)[/tex].
- In the second relation, the pair with [tex]\( x = 1.7 \)[/tex] and a counterpart [tex]\( y = -1.4 \)[/tex].
- Therefore, the value corresponding to [tex]\( x = 1.7 \)[/tex] in the second relation should match [tex]\( y = -1.4 \)[/tex] in the first relation, confirming that [tex]\( c = 3.2 \)[/tex].
4. Identify and fill in the missing values for [tex]\( d \)[/tex]:
- In the first relation, the pair at the fourth position is [tex]\((-0.2, 4.4)\)[/tex].
- In the second relation, the [tex]\( y \)[/tex] for [tex]\((-0.2)\)[/tex] is not known yet.
- Therefore, the position corresponding in the first relation should give us the value directly: since there's no existing pair just in the entered set information, we can't deduce [tex]\( d \)[/tex]. So, [tex]\( d = None \)[/tex].
5. Identify and fill in the missing values for [tex]\( e \)[/tex]:
- In the first relation, the pair at the fifth position is [tex]\((1.0, 5.0)\)[/tex].
- In the second relation, the pair with [tex]\( x = 5.0 \)[/tex] should have [tex]\( y \)[/tex] matching [tex]\( x's \)[/tex] partner in the first relation, confirming that [tex]\( e = 5.0 \)[/tex].
So, the values that make the two relations inverses of each other are:
- [tex]\( a = 3.2 \)[/tex]
- [tex]\( b = -3.1 \)[/tex]
- [tex]\( c = 3.2 \)[/tex]
- [tex]\( d = None \)[/tex]
- [tex]\( e = 5.0 \)[/tex]
Thus:
[tex]\[ \boxed{(a, b, c, d, e) = (3.2, -3.1, 3.2, \text{None}, 5.0)} \][/tex]
