Answer to Question #91466 in Algebra for Ra

Question

Which of the following statements are true? Please justify the answers.
1. For any two sets A and B, A∩Bᶜ=A\B.
2. The matrix [1 1 is singular.
0 0]
3. The contrapositive of '∃ y∈Z such that P(y) is true' is '∃ x∈Z such that P(x) is true'.
4. The system 2x-3y=1 and 6y-4x+2=0 has a unique solution.
5. If x, y∈C such that x²=y and y²=x, then x=y=1.

Answer 1

1. Relationship between relative and absolute complements:

$$A\setminus B=A\cap B^C$$

Therefore, the first statement is true.

2. A matrix is singular iff its determinant is 0.

$$\begin{vmatrix} 1 & 1 \\ 0 & 0 \end{vmatrix}=0-0=0$$

Therefore, the second statement is true.

3. A conditional statement has a contrapositive, a converse, and an inverse. The satement  '∃ y∈Z such that P(y) is true' is an existential statement and may have only negation.

Therefore, the third statement is not true.

4. Rewrite the system

$$2x-3y=1$$ $$4x-6y=2$$

The system has infinitely many solutions:

$$x={3\over 2}y+{1 \over 2}, y\in\R$$

Therefore, the fourth statement is not true.

5. Rewrite the system

$$x^2=y$$ $$y^2=x$$

Then

$$x=y^2 =x^4$$ $$x^4-x=0$$ $$x(x-1)(x^2+x+1)=0$$

$$x_1=0, x_2=1, x_3={-1\over 2}-{i\sqrt 3 \over 2}, x_4={-1\over 2}+{i\sqrt 3 \over 2}$$

$$y_1=0, y_2=1, y_3={-1\over 2}+{i\sqrt 3 \over 2}, y_4={-1\over 2}-{i\sqrt 3 \over 2}$$

Therefore, the fifth statement is not true.