The binary operation □ is defined as a □ b = ab + (a + b), where a and b are any two real numbers. The value of the identity element of this operation, defined as the number x such that a □ x = a, for any a, is_______.
0
Understanding Binary Operations and Identity Elements
A binary operation is a rule that combines two elements from a set to produce another element from the same set. In this question, the binary operation is defined as \(a \square b = ab + (a + b)\), where \(a\) and \(b\) are any two real numbers. We are asked to find the identity element for this specific operation.
What is an Identity Element in Binary Operations?
An identity element, often denoted by \(x\) or \(e\), is a special element within a set that, when combined with any other element \(a\) from the set using a given binary operation, leaves \(a\) unchanged. For an operation \(\square\), an element \(x\) is an identity element if for every element \(a\) in the set:
- \(a \square x = a\) (right identity)
- \(x \square a = a\) (left identity)
If an element satisfies both conditions, it is called the identity element.
Finding the Identity Element for the Operation \(a \square b = ab + (a + b)\)
To find the identity element, let's denote it as \(x\). According to the definition of a right identity element, we must have:
\[a \square x = a\]
Now, we substitute the given definition of the binary operation \(a \square b = ab + (a + b)\) into our equation. Replacing \(b\) with \(x\), we get:
\[ax + (a + x) = a\]
Next, we simplify the left side of the equation by removing the parentheses:
\[ax + a + x = a\]
Our goal is to solve for \(x\). To do this, we can subtract \(a\) from both sides of the equation:
\[ax + a + x - a = a - a\]
This simplifies to:
\[ax + x = 0\]
Now, we can factor out \(x\) from the terms on the left side of the equation:
\[x(a + 1) = 0\]
For this equation \(x(a + 1) = 0\) to hold true for any real number \(a\), the term \(x\) must be equal to zero. Let's consider why:
- If \(a+1 \neq 0\) (i.e., \(a \neq -1\)), then for the product \(x(a+1)\) to be zero, \(x\) must be \(0\).
- If \(a+1 = 0\) (i.e., \(a = -1\)), then the equation becomes \(x(0) = 0\), which is true for any \(x\). However, the definition of an identity element requires it to work for *any* \(a\), not just specific values. Therefore, \(x\) must be \(0\) to satisfy the condition for all values of \(a\) (including when \(a \neq -1\)).
Verification of the Identity Element
Let's verify if \(x = 0\) indeed works as the identity element for the given binary operation \(a \square b = ab + (a + b)\):
Substitute \(x = 0\) back into the definition of the operation \(a \square x\):
\[a \square 0 = a(0) + (a + 0)\]
Simplify the expression:
\[a \square 0 = 0 + a\]
\[a \square 0 = a\]
Since \(a \square 0 = a\), this confirms that \(0\) is the identity element for the given binary operation.
The value of the identity element of this operation is \(0\).
Important Questions from Instructions
Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows.
Statement:
I. All film stars are playback singers.
II. All film directors are film stars.
Conclusions:
I. All film directors are playback singers.
II. Some film stars are film directors.
All people in a certain are either ‘Knights’ or ‘Knaves’ and each person knows every other person’s identity. Knights NEVER lie, and knaves ALWAYS lie.
P says “Both of us are knights”, Q says “None of us are knaves”.
Which one of the following can be logically inferred from the above?Fact: If it rains, then the field is wet.
Read the following statements:
(i) It rains
(ii) The field is not wet
(iii) The field is wet
(iv) It did not rain
Which one of the options given below is NOT logically possible, based on the given fact?A smart city integrates all modes of transport, uses clean energy and promotes the sustainable use of resources. It also uses technology to ensure the safety and security of the city, something which critics argue, will lead to a surveillance state.
Which of the following can be logically inferred from the above paragraph?
(i) All smart cities encourage the formation of surveillance states.
(ii) Surveillance is an integral part of a smart city.
(iii) Sustainability and surveillance go hand in hand in a smart city.
(iv) There is a perception that smart cities promote surveillance.Social science disciplines were in existence in an amorphous form until the colonial period when they were intuitionalized. In varying degrees, they were intended to further the colonial interest. In the time of globalization and the economic rise of postcolonial countries like India, conventional ways of knowledge production have become obsolete.
Which of the following can be logically inferred from the above statements?
i) Social science disciplines have become obsolete.
ii) Social science disciplines had a pre – colonial origin
iii) Social science disciplines always promote colonialism
iv) Social science must maintain disciplinary boundaries.
