Here is the breakdown of the three mathematical problems in the image:

1. Consider the set $X := \mathbb{Q} \setminus \{1\}$ and an operation $\ast$ on $X$ as given below: $a \ast b = a + b - ab \quad \text{for all} \, a, b \in X.$ Show that $\ast$ is a binary operation on $X$. Find the identity element. Also, investigate the inverse of an element $x \in X$.

2. Let $G$ be a group and $c, d \in G$. Show that the equation $xc = d$ has a unique solution in $G$.

3. Let $a$ and $b$ be two elements of a group. If $\circ(a) = d$, then show that $\circ(a^{-1}) = \circ(bab^{-1})$.

Problem 1: Investigating the Binary Operation

• Binary operation: We need to verify that the operation $\ast$ is closed under the set $X = \mathbb{Q} \setminus \{1\}$. That is, for any $a, b \in X$, we check whether $a \ast b \in X$.
• Identity element: Find an element $e \in X$ such that for all $a \in X$, $a \ast e = e \ast a = a$.
• Inverse element: For each $x \in X$, find an element $y \in X$ such that $x \ast y = y \ast x = e$, where $e$ is the identity element.

Problem 2: Uniqueness of Solutions in a Group

• Since $G$ is a group, every element has an inverse, and the operation is associative. To show that the equation $xc = d$ has a unique solution, we will use properties of groups, such as the existence of inverses.

Problem 3: Group Element Relationship

• Given $\circ(a) = d$, we need to prove that $\circ(a^{-1}) = \circ(bab^{-1})$. This problem typically involves conjugation in a group and properties of inverse elements.

Let me know if you would like detailed solutions for each question or have any specific preferences!

Here are 5 related questions that can expand on this content:

1. What are the general conditions for a set and operation to form a binary operation?
2. How does the identity element in an operation relate to inverses?
3. What group properties ensure the uniqueness of solutions for group equations?
4. How does conjugation of elements in a group work, and how does it affect inverses?
5. Can you generalize the result of Problem 3 to non-abelian groups?

Tip: Always check the closure property when proving that an operation is binary on a set.