Use a Venn diagram to determine whether A > (B < C) 5
(A > B) < C for all sets A, B, and C.
In case you need to post the set notations, you can type
"& c a p ;" for ∩,
"& c u p ;" for ∪, and
"& s u b ;" for ⊂.
Do not type the double quotes nor the intervening spaces in the above expressions.
To determine whether A > (B < C) 5 is equivalent to (A > B) < C for all sets A, B, and C, let's first understand the meaning of the given expressions.
A > (B < C) is a comparison between the set A and the inequality operation (B < C) using the number 5 as the value. It indicates that A is greater than the result of the inequality operation (B < C) when the value 5 is used.
On the other hand, (A > B) < C is a comparison between the inequality operation (A > B) and the set C. It indicates that the result of the inequality operation (A > B) is less than the set C.
To determine their equivalence, we can construct a Venn diagram that represents the relationships between A, B, and C. Here's how we can do it:
1. Draw three overlapping circles, one for each set A, B, and C.
2. Place elements that belong to each set within their corresponding circles. Use different colors or markings to distinguish them.
3. Analyze each expression individually to determine which elements satisfy the given conditions.
Let's go step by step:
Step 1: Draw a Venn diagram with three circles labeled A, B, and C.
_______
/ \
| A |
\ _______ /
/ \
| B |
\ _______ /
/ \
| C |
\ _______ /
Step 2: Place the elements in each set A, B, and C. For example, let's say A = {1, 2, 3}, B = {2, 3, 4}, and C = {3, 4, 5}. Place the respective elements within the corresponding circles.
_______
/ 1,2,3 \
| A |
\ _______ /
/ 2,3,4 \
| B |
\ _______ /
/ 3,4,5 \
| C |
\ _______ /
Step 3: Analyze each expression individually.
a) For the expression A > (B < C) 5, start by evaluating (B < C) using the value 5:
B < C means each element in B should be less than each element in C. Therefore, (2 < 3), (3 < 4), and (4 < 5) are all true.
Next, we compare A with the result (B < C) using 5:
A > (B < C) 5 means that each element in A should be greater than the result of (B < C) when 5 is used. In our case, (1 > True), (2 > True), and (3 > True) are all false.
b) For the expression (A > B) < C, start by evaluating (A > B):
(1 > 2) is false, (2 > 3) is false, and (3 > 4) is false.
Next, we compare the result (A > B) with C:
(False < 3) is true, (False < 4) is true, and (False < 5) is true.
In conclusion, in our example scenario, A > (B < C) 5 and (A > B) < C are not equivalent. The first expression is false, while the second expression is true.