Question

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

• ⊆ means \is a subset of.\
• ⊂ means \is a proper subset of.\
• ⊈ means \is not a subset of.\
• ∅ is the empty set.

for each statement, decide if it is true or false.
(a) {c, d, h, m}⊈{c, d, h, m}
true false
(b) {1, 4}⊆{1, 2, 3, 4, ...}
true false
(c) ∅⊂{14, 17, 18, 19}
true false
(d) {p, q, r, s, t, u}⊂{q, s, u}
true false

Explanation:

Step1: Recall subset definition

A set is always a subset of itself. So $\{c,d,h,m\}
subseteq\{c,d,h,m\}$ is false.

Step2: Analyze set - inclusion

The set $\{1,4\}$ has elements that are also in the set of natural - numbers $\{1,2,3,4,\ldots\}$. So $\{1,4\}\subseteq\{1,2,3,4,\ldots\}$ is true.

Step3: Consider empty - set property

The empty set $\varnothing$ is a proper subset of any non - empty set. So $\varnothing\subset\{14,17,18,19\}$ is true.

Step4: Check proper - subset condition

The set $\{p,q,r,s,t,u\}$ has elements ($p$, $r$, $t$) that are not in $\{q,s,u\}$. So $\{p,q,r,s,t,u\}\subset\{q,s,u\}$ is false.

Answer:

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