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 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

a) False

b) True

c) True

d) True

e) True

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\\otin 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