Question

If R is the set of all integers with absolute value less than 10, A is its subset containing all natural numbers less than 10 and B is the set of all integer solutions of inequality 2x+5<9 that are less than 10 by absolute value (and therefore, it is also a subset of R), draw Venn diagram showing these sets. List elements of sets A, B, their union, and their intersection.

Answer 1

See explanation

Explanation:

1. R is the set of all integers with absolute value less than 10, thus

`R=\{a\in \mathbb{Z}\ :\ |a|<10 \}=\\ \\=\{-9,\ -8,\ -7,\ -6,\ -5,\ -4,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9\}`

2. A is its subset containing all natural numbers less than 10, thus

`A\subset R\\ \\A=\{b\in \mathbb{N}\ :\ b<10\}=\{1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9\}`

3. B is the set of all integer solutions of inequality 2x+5<9 that are less than 10 by absolute value (and therefore, it is also a subset of R). First, solve the inequality:

`2x+5<9\\ \\2x<9-5\\ \\2x<4\\ \\x<2`

Thus,

`B\subset R\\ \\B=\{c\in \mathbb{Z}\ :\ 2c+5<9,\ |c|<10\}=\{c\in \mathbb{Z}\ :\ c<2,\ |c|<10\}=\\ \\=\{-9,\ -8,\ -7,\ -6,\ -5,\ -4,\ -3,\ -2,\ -1,\ 0,\ 1\}`

See the diagram in attached diagram.

Note that

`A\cup B=R\\ \\A\cap B=\{1\}.`