Question

Answer the following true or false. Justify your answer.
(a) If A is a subset of B, and x∈B, then x∈A.
(b) The set {(x,y) ∈ R2 | x > 0 and x < 0} is empty.
(c) If A and B are square matrices, then AB is also square.
(d) A and B are subsets of a set S, then A∩B and A∪B are also subsets of S.
(e) For a matrix A, we define A^2 = AA.

Answer 1

Answer:

a) False

b) True

c) True

d) True

e) True

Step-by-step explanation:

a) Consider the sets $B=\{1,2,3,4,5,6\}$ and $A=\{4,5,6\}$. Observe that A is a subset of B and $1\in B$ but $1\notin A$

b) Any number different of 0 can be positive and negative simultaneusly. Then doesn't exist $x\in\mathbb{R}$ such that $x<0 and x>0$. Then the set $\{(x,y) \in \mathbb{R}^2 | x > 0\; \text{and}\; x < 0}$ is empty.

c) If the multiplication AB is defined and A and B are square matrices with A of size nxn, then B is the size nxn and the matrix AB is the size nxn.

d) Let A and B subsets of a set S. Since each element of A and B are in S then each element of $A\cup B$ is in S. Also, if $x\in A\cap B$, the $x\in A\subset S$ and $x\in B\subset S$ then $x\in S$. This shows that $A\cap B \subset S$.

e) By definition AA=A^2