Question

here are the meanings of some of the symbols that appear in the statements below

• $\subseteq$ means \is a subset of.\
• $\subset$ means \is a proper subset of.\
• $\

subseteq$ means \is not a subset of.\

• $\varnothing$ is the empty set.

for each statement, decide if it is true or false.
(a) $\\{1, 2, 3, 4, 5\\}\subset\\{2, 3, 5\\}$
\bigcirc true \bigcirc false
(b) $\\{12, 13\\}\subseteq\\{11, 12, 13, 14, \dots\\}$
\bigcirc true \bigcirc false
(c) $\\{g, h, j, m\\}\subseteq\varnothing$
\bigcirc true \bigcirc false
(d) $\\{22, 23, 24, 25, 29\\}\
subseteq\\{22, 23, 24, 25, 29\\}$
\bigcirc true \bigcirc false

Explanation:

Step1: Check subset for (a)

A set is a subset only if all its elements are in the other set. $\{1,2,3,4,5\}$ has elements 1,4 not in $\{2,3,5\}$, so $\{1,2,3,4,5\}
subseteq \{2,3,5\}$.

Step2: Check subset for (b)

All elements of $\{12,13\}$ (12 and 13) are in $\{11,12,13,14,...\}$, so $\{12,13\} \subseteq \{11,12,13,14,...\}$.

Step3: Check subset for (c)

The empty set $\emptyset$ has no elements. $\{g,h,j,m\}$ has elements, so it cannot be a subset of $\emptyset$.

Step4: Check non-subset for (d)

All elements of $\{22,23,24,25,29\}$ are in $\{22,23,24,25,29\}$, so it is a subset, meaning $\{22,23,24,25,29\}
subseteq \{22,23,24,25,29\}$ is false.

Answer:

(a) False
(b) True
(c) False
(d) False