问题1.查找一组I +通过A * B = A + B的所有A，B∈I +定义的所有正整数的单位元。
解决方案：
Let e be the identity element in I+ with respect to * such that
a * e = a = e * a, ∀ a ∈ I+
a * e = a and e * a = a, ∀ a ∈ I+
a + e = a and e + a = a, ∀ a ∈ I+
e = 0, ∀ a ∈ I+
Hence, 0 is the identity element in I+ with respect to *.
问题2。在所有有理数的集合中找到恒等式元素，除了–关于a定义的*的* 1 = b = a + b + ab
解决方案：
Let e be the identity element in I+ with respect to * such that
a * e = a = e * a, ∀ a ∈ Q – {-1}
a * e = a and e * a = a, ∀ a ∈ Q – {-1}
a + e + ae = a and e + a + ea = a, ∀ a ∈ Q – {-1}
e + ae = 0 and e + ea = 0, ∀ a ∈ Q – {-1}
e (1 + a) = 0 and e (1 + a) = 0, ∀ a ∈ Q – {-1}
e = 0, ∀ a ∈ Q – {-1} [because a not equal to -1]
Hence, 0 is the identity element in Q – {-1} with respect to *.
问题3.如果集合Z上的二进制运算*由a * b = a + b – 5定义，则找到关于*的恒等元素。
解决方案：
We are given the binary operator * defined on Z as
a*b = a + b – 5 for all a, b ∈ Q
Let e be the identity elements with respect to *
Then, a*e = e*a = a [By identity property]
⇒ a + e – 5 = a
⇒ e = 5
Therefore, the required identity element with respect to * is 5.
问题4.在集合Z的整数上，如果二进制运算*由a * b = a + b + 2定义，则查找标识元素。
解决方案：
The binary operator * is defined on Z, and is given by
a*b = a + b +2 for all a, b ∈ Z.
Let a ∈ Z and e ∈ Z be the identity element with respect to *, then
a*e = e*a = a [By identity property]
⇒ a + e + 2 = a
⇒ e = -2 ∈ Z
Therefore, the identity element with respect to * is -2.