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.
1
Expert's answer
2019-07-09T12:08:38-0400
1. Relationship between relative and absolute complements:
A∖B=A∩BC
Therefore, the first statement is true.
2. A matrix is singular iff its determinant is 0.
∣∣1010∣∣=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.
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