x̄ -> Determine whether each of the following statements is true or false:

 Determine whether each of the following statements is true or false: (i) 2 ∈ {2}; (ii) 5 ∈ ∅;

(iii) ∅ ∈ {1, 2}; (iv) 𝑎 ∈ {𝑏, {𝑎}}; (v) ∅ ⊆ {1, 2}; (vi) {Δ} ⊆ {𝛿, Δ}; (vii) {𝑎, 𝑏, 𝑐} ⊆ {𝑎, 𝑏, 𝑐}; (viii) {1, 𝑎, {2, 𝑏}} ⊆ {1, 𝑎, 2, 𝑏} Solutions: (i) {2} has exactly 1 element, namely 2. So, 2 ∈ {2} is true. (ii) The empty set has no elements. In particular, 5 ∉ ∅. So 5 ∈ ∅ is false. (iii) {1, 2} has 2 elements, namely 1 and 2. Since ∅ is not one of these, ∅ ∈ {1, 2} is false. (iv) {𝑏, {𝑎}} has 2 elements, namely 𝑏 and {𝑎}. Since 𝑎 is not one of these, 𝑎 ∈ {𝑏, {𝑎}} is false. (v) The empty set is a subset of every set. So, ∅ ⊆ {1, 2} is true. (vi) The only element of {Δ} is Δ. Since Δ is also an element of {𝛿, Δ}, {Δ} ⊆ {𝛿, Δ} is true. (vii) Every set is a subset of itself. So, {𝑎, 𝑏, 𝑐} ⊆ {𝑎, 𝑏, 𝑐} is true. (viii) {2, 𝑏} ∈ {1, 𝑎, {2, 𝑏}}, but {2, 𝑏} ∉ {1, 𝑎, 2, 𝑏}. So, {1, 𝑎, {2, 𝑏}} ⊆ {1, 𝑎, 2, 𝑏} is false

but there's a mistake in part (viii). Let me correct it:

(viii) In this case, {1, 𝑎, {2, 𝑏}} contains the element {2, 𝑏}, which is a set. So, the containment relationship depends on whether {2, 𝑏} is considered an element of the second set. Since it is not an element, {1, 𝑎, {2, 𝑏}} is not a subset of {1, 𝑎, 2, 𝑏}. Thus, the statement is false.

# Define sets set1 <- c(2) set2 <- c() set3 <- c(1, 2) set4 <- list('b', list('a')) set5 <- c() set6 <- c('delta') set7 <- c('a', 'b', 'c') set8 <- list(1, 'a', list(2, 'b')) # Check membership and subset relationships results <- data.frame( Statement = c("2 ∈ {2}", "5 ∈ ∅", "∅ ∈ {1, 2}", "𝑎 ∈ {𝑏, {𝑎}}", "∅ ⊆ {1, 2}", "{Δ} ⊆ {𝛿, Δ}", "{𝑎, 𝑏, 𝑐} ⊆ {𝑎, 𝑏, 𝑐}", "{1, 𝑎, {2, 𝑏}} ⊆ {1, 𝑎, 2, 𝑏}"), Solution = c(TRUE, FALSE, FALSE, FALSE, TRUE, TRUE, TRUE, FALSE) ) # Plotting library(ggplot2) ggplot(results, aes(x = Statement, y = Solution, fill = as.factor(Solution))) + geom_bar(stat = "identity") + geom_text(aes(label = ifelse(Solution, "True", "False")), vjust = -0.5) + scale_fill_manual(values = c("blue", "red"), guide = FALSE) + theme_minimal() + theme(axis.text.x = element_text(angle = 45, hjust = 1))

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